On the Higher Frobenius
by Allen Yuan
Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2020 © Massachusetts Institute of Technology 2020. All rights reserved.
Author...... Department of Mathematics April 30, 2020
Certified by ...... Jacob Lurie Professor of Mathematics Thesis Supervisor
Accepted by...... Davesh Maulik Chairman, Department Committee on Graduate Theses 2 On the Higher Frobenius by Allen Yuan
Submitted to the Department of Mathematics on April 30, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Abstract Given a homotopy invariant of a space, one can ask how much of the space can be recovered from that invariant. This question was first addressed in work of Quillen and Sullivan on rational homotopy theory in the 1960’s and in work of Dwyer- Hopkins and Mandell on 푝-adic homotopy theory in the 1990’s. In this thesis, we describe a way to unify these ideas and recover a space in its entirety, rather than up to an approximation. The approach is centered around the study of the higher Frobenius map. First defined by Nikolaus and Scholze, the higher Frobenius map generalizes to E∞-ring spectra the classical Frobenius endomorphism for rings in characteristic 푝. Our main result is that there is an action of the circle group on (a certain subcategory of) 푝-complete E∞-rings whose monodromy is the higher Frobenius. Using this circle action, we give a fully faithful model for a simply connected finite complex 푋 in terms of Frobenius-fixed E∞-rings.
Thesis Supervisor: Jacob Lurie Title: Professor of Mathematics
3 4 Acknowledgments
First and foremost, I would like to thank my advisor, Jacob Lurie. His mathematics has continually inspired me and shaped my thinking; equally importantly, this thesis would not exist without his unending patience, intellectual generosity, and constant support throughout my time in graduate school. Special thanks goes also to Haynes Miller, for taking a confused undergraduate into the Kan seminar and spending countless hours in his office teaching me. Thank you to all my teachers and mentors throughout the years; while there’s no way to properly acknowledge all of them here, particular thanks are due to Yum-Tong Siu, Joe Harris, Peter Kronheimer for starting me off right, Emily Riehl, Hiro Tanaka, and Eric Peterson for going above and beyond, Ken Ono for his energetic support, and Mike Hopkins for his kindness, insights, and advice. My journey would not be without my wonderful friends, colleagues, and collab- orators; thanks to Hood Chatham, Andy Senger, Peter Haine, Robert Burklund, Elden Elmanto, Gijs Heuts, Lukas Brantner, Akhil Mathew, Amol Aggarwal, Carl Lian, and many many others. I’m especially grateful to Arpon Raksit for always showing me how to do things the right way, and to Jeremy Hahn for his friend- ship, mentorship and fruitful collaboration. And to Kevin, Joy, Julia, Spencer, and Stella – thanks for all the good times, and may this just be the beginning. The material in this thesis additionally benefitted from conversations with Clark Barwick and Denis Nardin, and was built on the beautiful ideas of Thomas Nikolaus, whom I would like to heartily thank for his generosity and encourage- ment. This work was also supported in part by the NSF under Grant DGE- 1122374. I would like to thank my parents for their love, nurturing, and support, and for all the things that one only appreciates years later. Thank you as well to Susu, for always caring for and challenging me. Finally, to In Young – your unconditional love and encouragement pulls me through the hurdles. You are amazing and thank you for everything.
5 6 Contents
1 Introduction 9 1.1 Summary of results ...... 10 1.2 Outline of thesis ...... 11 1.3 Organization ...... 20
2 Equivariant Stable Homotopy Theory 23 2.1 Global equivariance ...... 23 2.2 Formal constructions around Glo+ ...... 27 2.3 Borel equivariant spectra and proper Tate constructions ...... 29
3 The Integral Frobenius Action 33
3.1 The E∞-Frobenius ...... 34 3.2 Generalized Frobenius and canonical maps ...... 35 3.3 Global algebras ...... 38 3.4 Borel global algebras ...... 43
4 Partial Algebraic 퐾-theory 51
4.1 Partial 퐾-theory via the 푆∙-construction ...... 52 4.2 Partial K-theory via the 푄-construction ...... 58
5 The Partial 퐾-theory of F푝 63 5.1 Computing partial 퐾-theory ...... 64 5.2 The partial 퐾-theory of F푝 ...... 70
6 The 푝-complete Frobenius and the Action of 퐵 Z≥0 79 6.1 퐹푝-stable E∞-rings ...... 80 6.2 The action of 퐵 Z≥0 on 퐹푝-stable E∞-rings ...... 84
7 1 6.3 Perfect E∞-rings and the action of 푆 ...... 86
7 Integral Models for Spaces 89 7.1 푝-adic homotopy theory over the sphere ...... 89 7.2 Integral models for spaces ...... 94 7.3 Further extensions and questions ...... 96
A Generalities on Producing Actions 99
8 Chapter 1
Introduction
In algebra, one often studies questions over the integers by separately understand- ing them over Q and after completion at each prime 푝. The analog of this idea in topology, due to Sullivan [44], is that any space 푋 can be approximated by ∧ its rationalization 푋Q and its 푝-completions 푋푝 . These approximations, in turn, can be understood in terms of their algebras of cochains. Work of Sullivan (and
Quillen in a dual setting) shows that for sufficiently nice 푋, the rationalization 푋Q is captured completely by a commutative differential graded algebra (cdga) which is quasi-isomorphic to 퐶*(푋, Q). More precisely:
Theorem 1.0.1 (Sullivan [43], Quillen [36]). The assignment 푋 ↦→ 퐶*(푋, Q) de- termines a fully faithful functor from the ∞-category of simply connected rational spaces of finite type to the ∞-category CAlgQ of rational cdgas.
In the 푝-adic case, Mandell proves an analogous result with cdgas over Q re- placed by E∞-algebras over F푝:
* Theorem 1.0.2 (Mandell [31]). The assignment 푋 ↦→ 퐶 (푋, F푝) determines a fully faithful functor from the ∞-category of simply connected 푝-complete spaces ∞ CAlg of finite type to the -category F푝 of E∞-algebras over F푝.
The next natural question is whether a similar cochain model for spaces exists integrally.
Warning 1.0.3. Mandell [32] shows that the assignment 푋 ↦→ 퐶*(푋; Z) deter- mines a functor from spaces to E∞-algebras over Z that is faithful but not full.
9 Let us illustrate the difficulty in producing such an integral model: Sullivan showed that any sufficiently nice space 푋 can be recovered from its rationalization and 푝-completions together with the additional data of maps
∧ 푋Q → (푋푝 )Q for each prime 푝 via a homotopy pullback square
∏︁ ∧ 푋 푋푝 푝
∏︁ ∧ 푋Q (푋푝 )Q. 푝
One might hope to construct the desired integral model for spaces by assembling ∧ the above cochain models, for 푋Q and each 푋푝 , via an analogous procedure. * However, it is unclear how this assembly would work. The E∞-algebras 퐶 (푋; Q) * and 퐶 (푋; F푝) are defined over fields of different characteristic, so there isno obvious way to compare them.
1.1 Summary of results
In this paper, we give an integral cochain model for spaces. We accomplish this by producing a Frobenius action on certain E∞-ring spectra and modeling spaces in terms of the fixed points of this action. We now state our main theorems informally, with more precise statements to follow later in this introduction.
Theorem A (Frobenius Action). For each prime 푝, the Nikolaus–Scholze Frobenius determines an action of the monoidal category 퐵 Z≥0 on (a certain subcategory of) 푝-complete E∞-rings. We may then consider the homotopy fixed points for this Frobenius action, which we call 푝-Frobenius fixed E∞-rings. We will explain that a 푝-Frobenius fixed E∞-ring is the data of an E∞-ring 퐴 equipped with a particular sequence of homotopies, the first of which is a trivialization of its Frobenius endomorphism (cf. Warning 1.2.5).
10 We use this notion to give a new 푝-adic model for a space 푋, which we will assume to be a simply connected finite complex. Theorem B (푝-adic Model). Let 푝 be a prime. Then:
∧ 푋 • The E∞-algebra (푆푝 ) of cochains on 푋 with values in the 푝-complete sphere is naturally a 푝-Frobenius fixed E∞-ring.
∧ ∧ 푋 • The data of the space 푋푝 is captured completely by (푆푝 ) as a 푝-Frobenius fixed E∞-ring. These 푝-adic models are more amenable to assembly into an integral model. In 푋 particular, Theorem B implies that the E∞-ring 푆 of sphere-valued cochains on 푋 (i.e., its Spanier-Whitehead dual) is what we will call Frobenius fixed, meaning that for each prime 푝, its 푝-completion is 푝-Frobenius fixed. These Frobenius fixed E∞-rings are our integral model for spaces. Theorem C (Integral Model). The data of the space 푋 is captured completely by 푋 its Spanier-Whitehead dual 푆 as a Frobenius fixed E∞-ring. We now proceed to a more detailed outline of the paper.
1.2 Outline of thesis
1.2.1 Integral models for spaces and the Frobenius
Let 푝 be a prime and 푋 be a simply connected 푝-complete space of finite type. * Mandell’s theorem (Theorem 1.0.2) asserts that the functor 푋 ↦→ 퐶 (푋; F푝) is fully faithful, so 푋 can be recovered as the mapping space
푋 ≃ CAlg (퐶*(푋, ), ). F푝 F푝 F푝
* On the other hand, the assignment 푋 ↦→ 퐶 (푋, F푝) fails to be fully faithful: one has that
CAlg (퐶*(푋, ), ) ≃ CAlg (︀퐶*(푋, ), )︀ℎ Z F푝 F푝 F푝 F푝 F푝 F푝 ≃ CAlg (퐶*(푋, ), )ℎ Z ≃ 푋ℎ Z ≃ ℒ푋 F푝 F푝 F푝
CAlg with Z acting on F푝 by Frobenius [31]. If this functor to F푝 were fully faithful, the result of this calculation would be 푋; we see that the extra free loop space has
11 to do with a failure to account for the Frobenius. In analogy to classical algebra, one might hope that any E∞-algebra over F푝 has a natural Frobenius map, and that the 푆1-action on ℒ푋 = Hom(푆1, 푋) arises intrinsically on the left-hand side 푆1 ∞ CAlg from an -action on the -category F푝 whose monodromy on any object is Frobenius. Then, taking fixed points for 푆1 would “undo” the free loop space that appears in the above calculation. In fact, there is a candidate for this Frobenius map, first defined by Nikolaus- Scholze [34]. For any E∞-ring 퐴 and prime 푝, they constructed a natural map of 푡퐶푝 rings 휙퐴 : 퐴 → 퐴 which we call the E∞-Frobenius (or simply the Frobenius). This map takes values not in the ring 퐴 itself, but in the 퐶푝-Tate cohomology of 퐴. Accordingly, 휙퐴 is not generally an endomorphism; it is the E∞ analog of the 푝th power map 푅 → 푅/푝 for an ordinary ring 푅 (cf. Example 3.1.4). In particular, 푡퐶푝 the E∞-Frobenius is not an endomorphism for the E∞-ring F푝 because F푝 is not equivalent to F푝, so the conjectural picture above is not correct as stated. However, the main thesis of this paper is that one can realize this picture by working not with F푝-algebras, but with algebras over the 푝-complete sphere.
Example 1.2.1. Any E∞-ring spectrum 퐴 admits another canonical ring map 푡퐶푝 can퐴 : 퐴 → 퐴 by the composite
퐴 → 퐴ℎ퐶푝 → 퐴푡퐶푝 of restriction to group cohomology of 퐶푝 with projection to Tate cohomology. ∧ When 퐴 = 푆푝 , the 푝-complete sphere, Lin’s theorem [25] asserts that this canonical map is an equivalence. Thus, the Frobenius can be regarded as an endomorphism ∧ of 푆푝 .
퐹 Motivated by this example, we extract a full subcategory CAlg푝 of 푝-complete E∞-rings which we call 퐹푝-stable (Definition 6.1.1); a 푝-complete E∞-ring 퐴 is 퐹푝-stable if it satisfies a condition slightly stronger than requiring can퐴 to be an equivalence. We will see that any E∞-ring which is finite over the 푝-complete sphere is 퐹푝-stable.
The Frobenius can be regarded as a natural endomorphism of any 퐹푝-stable ring, so one can define a functor
퐹 퐹 Φ: 퐵 Z≥0 → Fun(CAlg푝 , CAlg푝 )
12 sending the unique object of 퐵 Z≥0 to the identity functor and the morphism 1 ∈ Z≥0 to the Frobenius, considered as a natural transformation id → id.
Question. Does Φ lift to a monoidal functor and thus define an action of 퐵 Z≥0 퐹 on the ∞-category CAlg푝 ?
Remark 1.2.2. This question is due to Jacob Lurie and Thomas Nikolaus, who formulated it in the dual setting of 푝-complete bounded below E∞-coalgebras. We address this closely related case in upcoming work [47].
Our first main theorem answers this question to the affirmative:
퐹 Theorem A (Frobenius Action). The ∞-category CAlg푝 of 퐹푝-stable E∞-rings admits an action of 퐵 Z≥0 for which 1 ∈ Z≥0 acts by the Frobenius.
We will elaborate on Theorem A and its proof starting in §1.2.2 below. For now, we return to the application in mind with a straightforward corollary of Theorem A.
∘ perf 퐹 Theorem A . Let CAlg푝 ⊂ CAlg푝 denote the full subcategory of 퐹푝-stable perf E∞-rings for which the Frobenius acts invertibly. Then the ∞-category CAlg푝 admits an action of 푆1 whose monodromy induces the Frobenius automorphism on each object.
perf We refer to E∞-rings 퐴 ∈ CAlg푝 as 푝-perfect.
∧ Example 1.2.3. The 푝-complete sphere 푆푝 is 푝-perfect because the Frobenius is ∧ a ring map and 푆푝 is the initial 푝-complete E∞-ring. One can deduce from this ∧ 푋 that for any finite space 푋, the E∞-algebra of cochains (푆푝 ) is 푝-perfect.
As suggested above, Theorem A∘ allows one to model 푝-adic unstable homotopy theory in terms of algebras over the 푝-complete sphere.
휙=1 perf ℎ푆1 Definition 1.2.4. Let CAlg푝 := (CAlg푝 ) denote the ∞-category of homo- topy fixed points under the 푆1-action of Theorem A∘. We will refer to objects of 휙=1 휙=1 CAlg푝 as 푝-Frobenius fixed E∞-rings, and we will refer to a lift 퐴휙=1 ∈ CAlg푝 of a 푝-perfect E∞-ring 퐴 to a 푝-Frobenius fixed E∞-ring as an 퐹푝-trivialization for 퐴.
13 fsc Let 풮푝 denote the full subcategory of spaces which are homotopy equivalent to the 푝-completion of a simply connected finite complex. The following theorem fsc gives a model for 풮푝 in terms of E∞-algebras and the Frobenius action of Theorem A.
fsc op ∧ 푋 Theorem B 푝 . (풮 ) → CAlg ∧ 푋 ↦→ (푆 ) ( -adic Model) The functor 푝 푆푝 given by 푝 lifts to a fully faithful functor
∧ (−) fsc op 휙=1 (푆푝 )휙=1 :(풮푝 ) → CAlg푝 .
Theorem B asserts that for any simply connected finite 푝-complete space 푋, ∧ 푋 the algebra of cochains (푆푝 ) can be equipped with a canonical 퐹푝-trivialization, ∧ 푋 and that 푋 is captured completely by the E∞-ring (푆푝 ) together with its 퐹푝- trivialization.
Warning 1.2.5. An 퐹푝-trivialization is not simply a trivialization of the E∞- Frobenius map. To illustrate this, note first that the 푆1 action of Theorem A∘ perf,≃ 1 allows one to construct a fibration 푞 : (CAlg푝 )ℎ푆1 → 퐵푆 of spaces (the action groupoid). This begets the outer pullback square in the diagram:
perf,≃ perf,≃ CAlg푝 (CAlg푝 )ℎ푆1
푞
* C푃 1 ··· 퐵푆1 ≃ C푃 ∞
The data of a 푝-Frobenius fixed algebra is the data of a section of 푞. One can view such a section as being built cell-by-cell along the cell decomposition ∞ of C푃 . A section of 푞 over the point is the data of a 푝-perfect E∞-ring 퐴. Extending this section to a section over C푃 1 is just the data of a trivialization of the Frobenius map on 퐴. However, to promote this to an 퐹푝-trivialization of 퐴, one needs to further extend this section over all of C푃 ∞; this requires one to specify an additional homotopy for each additional cell of C푃 ∞. As promised, this model of 푝-complete spaces is more compatible with the CAlg ∞ CAlg휙=1 푝 rational model of Theorem 1.0.1; unlike F푝 , the -category 푝 of - CAlg Frobenius fixed algebras admits an obvious functor to Q푝 by forgetting the ∧ 퐹푝-trivialization and extending scalars along the map 푆푝 → Q푝. This allows our
14 푝-adic model to be compared to Sullivan’s rational model for spaces. We now describe the resulting integral model for spaces.
Definition 1.2.6. We will say that an E∞-ring 퐴 is perfect if for each prime 푝, ∧ perf its 푝-completion 퐴푝 is 푝-perfect. Let CAlg ⊂ CAlg be the full subcategory 휙=1 of perfect E∞-rings. Additionally, let the ∞-category CAlg of Frobenius fixed E∞-rings be defined by the pullback square
휙=1 ∏︁ 휙=1 CAlg CAlg푝 p 푝
∏︀ ∧ perf (−)푝 ∏︁ perf CAlg CAlg푝 . 푝
Informally, a Frobenius fixed E∞-ring is a perfect E∞-ring 퐴 equipped with an ∧ 퐹푝-trivialization of its 푝-completion 퐴푝 for each prime 푝. Let 풮fsc denote the full subcategory of spaces which are homotopy equivalent to a simply connected finite complex. We obtain the following model for 풮fsc as a corollary of Theorem B and Theorem 1.0.1: Theorem C (Integral Model). The functor (풮fsc)op → CAlg given by 푋 ↦→ 푆푋 lifts to a fully faithful functor
(−) fsc op 휙=1 푆휙=1 :(풮 ) → CAlg .
Theorem C asserts that for any simply connected finite space 푋, the algebra 푋 푆 of spherical cochains can be canonically promoted to a Frobenius fixed E∞-ring 푋 푋 푆휙=1 and that the data of 푋 is completely specified by 푆휙=1. Remark 1.2.7. Unlike Theorem 1.0.2, this spherical model only works for finite simply connected spaces (see also §7.3). However, it is the case that any simply connected space is a filtered colimit of finite ones, so one could model thewhole ∞-category of simply connected spaces by pro-objects in Frobenius fixed E∞-rings.
1.2.2 The Frobenius action
We return to the main technical focus of this paper, which is the development of the Frobenius action of Theorem A.
15 Remark 1.2.8. The analog of Theorem A in classical algebra is easy. Letting CAlg♡ F푝 denote the category of discrete commutative F푝-algebras, we can define a functor Φ♡ : 퐵 → Fun(CAlg♡ , CAlg♡ ) Z≥0 F푝 F푝 by sending 1 ∈ Z≥0 to the Frobenius endomorphism for discrete F푝-algebras. This functor is monoidal if and only the corresponding map on centers (i.e., endomor- phisms of the unit object)
Z≥0 → End(idCAlg♡ ) F푝 is a map of commutative monoids. This is automatic because the category of commutative monoids is a full subcategory of the category of monoids; in algebra, commutativity is a property.
End(id 퐹 ) This argument breaks down in homotopy theory. The space CAlg푝 is naturally E2-monoidal because it arises as endomorphisms of the unit object in 퐹 퐹 ∞ Fun(CAlg , CAlg ) 휙 ∈ End(id 퐹 ) the monoidal -category 푝 푝 . The Frobenius CAlg푝 determines a map of E1-spaces
→ End(id 퐹 ) Z≥0 CAlg푝 because Z≥0 is free as an E1-monoid, and one needs to show that it can be promoted to a map of E2-spaces. This is no longer automatic and the content of Theorem A is that this is possible.
휓 , 휓 ∈ End(id 퐹 ), Theorem A has the following concrete consequence: for any 1 2 CAlg푝 the E2-structure provides a homotopy 휓1휓2 ∼ 휓2휓1. This induces an action of the 푛 braid group on 푛 strands 휙 : id 퐹 → id 퐹 . on the natural transformation CAlg푝 CAlg푝 Theorem A asserts that this action is trivial, which can be thought of heuristically as saying that the Frobenius “commutes with itself.”
The majority of the paper is dedicated to proving Theorem A. The proof pro- ceeds in two steps, which we believe may be of independent interest and which we outline in §1.2.3 and §1.2.4, respectively. Roughly, the first step is to use equiv- ariant stable homotopy theory to produce a “lax” version of the Frobenius action 퐹 which exists for all E∞-rings, without 푝-completing or restricting to CAlg푝 (The- orem A♮). The second step is to use a “pre-group-completed” variant of algebraic 퐾-theory which we call partial 퐾-theory to descend this to an action of 퐵 Z≥0 on
16 퐹 the full subcategory CAlg푝 ⊂ CAlg.
1.2.3 Frobenius for genuine globally equivariant rings
The Frobenius action of Theorem A arises by observing that one can associate a Frobenius operator to every finite group 퐺 and then carefully studying the interac- tion between these operators. We describe this using equivariant stable homotopy theory. In §3, we construct an ∞-category CAlgGlo of global algebras (Definition 3.3.2), which are roughly E∞-rings 퐴 together with the additional structure, for each finite group 퐺, of a genuine 퐺-equivariant multiplication map 푁 퐺퐴 → triv퐺 퐴 from the norm of 퐴 to the ring 퐴 with trivial 퐺-action. The genuine multiplica- tions on a global algebra can be thought of as lifting various composites of the E∞-Frobenius to endomorphisms of 퐴 (cf. Example 3.3.1). As such, global al- gebras admit natural Frobenius endomorphisms for each 퐺, and we denote the associated natural transformation by 휙퐺 : id → id. The endomorphisms 휙퐺 exhibit interesting functorialities in the group 퐺: we observe that the various Frobenius operators assemble into an action of a symmet- ric monoidal 1-category 풬 (roughly, Quillen’s 푄-construction on finite groups, cf. Definition 3.2.5), whose objects are finite groups and whose morphisms from 퐻 to 퐺 are isomorphism classes of spans (퐻 퐾 ˓→ 퐺). Namely, in Theorem 3.3.6, we show that 풬 acts on the ∞-category CAlgGlo in a way such that each object 퐺 in 풬 acts trivially and the span (* * ˓→ 퐺) is sent to 휙 . The action of 풬 on the ∞-category of global algebras does not directly induce an action on the ∞-category of E∞-rings. A global algebra is, in general, more data Glo than an E∞-ring. However, we explain that the ∞-categories CAlg and CAlg are related via the notion of a Borel global algebra (Definition 3.4.5, Theorem 푡퐶푝 3.4.8). We proceed to deduce that the E∞-Frobenius 퐴 → 퐴 extends to an oplax action of 풬 on the ∞-category CAlg of E∞-ring spectra; this oplax action is unwound explicitly in §3.2 and is the content of Theorem 3.2.7, stated here in rough form:
Theorem A♮ (Integral Frobenius Action). There is an oplax monoidal functor
풬 → Fun(CAlg, CAlg)
푡퐶푝 extending the E∞-Frobenius in the sense that 퐶푝 ∈ 풬 is sent to (−) and the
17 푡퐶푝 span (* * ˓→ 퐶푝) acts by the Frobenius id → (−) .
♮ We emphasize that Theorem A applies to all E∞-rings and finite groups, and in particular does not require 푝-completion or passing to a subcategory of rings. It can be seen as describing the relationship between various “stable power operations” on E∞-rings (cf. Remark 3.2.9).
1.2.4 Partial Algebraic 퐾-theory
Recall that the 푄-construction is a device introduced by Quillen in order to define higher algebraic 퐾-theory. In this light, the Frobenius acting through 풬 can be seen as articulating that the action of Frobenius is “퐾-theoretic” or “additive in exact sequences” in the sense that for a short exact sequence 퐺′ → 퐺 → 퐺′′ of groups, the Frobenius for 퐺 is equivalent to the Frobenius for 퐺′ followed by the Frobenius for 퐺′′. Theorem A follows from Theorem A♮ by making this idea precise. Fixing a prime 푝, we restrict the oplax action of Theorem A♮ to the full sub- 퐹 categories CAlg푝 ⊂ CAlg and 푄VectF푝 ⊂ 풬 (spanned by the elementary abelian 퐹 푝-groups) to obtain a (strong) action of 푄VectF푝 on CAlg푝 . Recall the following computation of Quillen regarding 푄VectF푝 :
Theorem 1.2.9 (Quillen [37]). The natural map
Ω|푄VectF푝 | =: 퐾(F푝) → 휋0퐾(F푝) ≃ Z induces an isomorphism in F푝-homology.
Motivated by this computation, one might hope that our action of 푄VectF푝 extends to an action of the underlying E∞-space |푄VectF푝 |. Since |푄VectF푝 | ≃ 1 1 퐵퐾(F푝) is 푝-adically equivalent to 퐵 Z ≃ 푆 , the resulting 푆 action would restrict 1 to the desired action of 퐵 Z≥0. Unfortunately, this is impossible because 푆 is group 퐹 complete, and the Frobenius on CAlg푝 does not necessarily act by equivalences. We overcome this difficulty in Chapter 4 by introducing a non-group-complete variant of algebraic 퐾-theory which we call partial 퐾-theory (Definition 4.1.5). This construction takes a Waldhausen category (more generally, ∞-category [7]) part 풞 and produces an E∞-space 퐾 (풞) such that:
part gp • There is a canonical equivalence of E∞-spaces 퐾 (풞) ≃ 퐾(풞).
18 part • The monoid 휋0(퐾 (풞)) is freely generated by the objects of 풞 subject to the relation [퐴]+[퐶] = [퐵] for every short exact sequence 0 → 퐴 → 퐵 → 퐶 → 0. We show that many statements in algebraic 퐾-theory have analogs in partial part 퐾-theory. For instance, while the definition of 퐾 (풞) uses Waldhausen’s 푆∙- construction, we give an alternate construction in the case of an exact category 풞 involving Quillen’s 푄-construction and show that the two definitions coincide (Theorem 4.2.3). It follows from this latter construction of partial 퐾-theory that the action of 퐹 part 푄VectF푝 on CAlg푝 descends to an action of the monoidal ∞-category 퐵퐾 (F푝) 퐹 on CAlg푝 . The 퐵 Z≥0 action of Theorem A arises by combining this action of part 퐵퐾 (F푝) with the following partial 퐾-theory analog of Theorem 1.2.9: Theorem 5.0.2. The natural map
part part 퐾 (F푝) → 휋0퐾 (F푝) ≃ Z≥0 induces an isomorphism in F푝-homology. In addition to proving Theorem A, our methods also imply the following as- sertions about global algebras, which may be of independent interest: Theorem 6.2.2.
1. The ∞-category CAlgGlo of global algebras admits an action of the monoidal ∞-category 퐵퐾part(Z) for which an abelian group 퐺 acts by the Frobenius 휙퐺 : id → id.
Glo Glo 2. The full subcategory CAlg푝 ⊂ CAlg of global algebras with 푝-complete underlying E∞-ring admits an action of 퐵 Z≥0 for which 1 ∈ Z≥0 acts by the Frobenius (for the group 퐶푝). Remark 1.2.10. The notion of a global algebra is not new; they have appeared in the literature as the normed algebras of [4] and are closely related to Schwede’s ultra-commutative monoids [40].
1.2.5 Relationship to other work
There has been much previous work on understanding spaces via chain and cochain functors. The rational case is due to Quillen [36] and Sullivan [43]. Recent work
19 of Heuts [17] and Behrens-Rezk [10] extends this rational picture to the setting of 푣푛-periodic spaces. In another vein, Goerss [15] gives a model for 푝-local spaces in terms of simplicial coalgebras, and Kriz [23] for 푝-complete spaces in terms of cosimplicial algebras. Our work is closest in spirit to work of Dwyer-Hopkins (unpublished) and Man- dell [31], who study 푝-complete spaces via E∞-algebras over F푝. In [32], Mandell further explains that the functor of integral cochains from (nice enough) spaces to E∞-algebras over Z is faithful but not full. This paper essentially realizes a program due to Thomas Nikolaus, in the setting of coalgebras, for making a fully faithful model. The idea for this starts with unpublished work of Mandell which lifts 푝-adic homotopy theory from F푝 to the spherical Witt vectors of F푝. Then, it is an insight of Nikolaus (in the setting of coalgebras) that having a homotopy coherent Frobenius action allows one to descend to a statement over the sphere. Our contribution is to produce this Frobenius action, and to make sense of this picture in the dual setting of algebras. We also acknowledge that Nikolaus has concurrent work in the dual setting of 푝-complete perfect E∞-coalgebras. In forthcoming work [47], we show how our methods adapt to the setting of coalgebras to produce a 퐵 Z≥0-action on all 푝-complete bounded below E∞-coalgebras. We will also (at least conjecturally) describe the Goodwillie filtration in that setting, relating our model to the Tate coalgebras of Heuts [16, §6.4].
1.3 Organization
In §2, we record the requisite preliminaries on equivariant stable homotopy theory. Then in §3, we introduce the E∞-Frobenius and use the results of §2 to prove Theorem A♮ (as Theorem 3.2.7). In §4, we introduce and develop basic results about partial 퐾-theory. Then in §5, we present our results on the partial 퐾-theory of F푝 (Theorem 5.0.2). In §6, we combine the previous results to prove Theorem A. Then in §7, we apply Theorem A to prove Theorems B and C about E∞-algebra models for spaces. Finally, in the appendix, we settle certain technical constructions needed in §3.
20 1.3.1 Notations and conventions
We use the following notations and conventions throughout:
• We use F푝 instead of 퐻 F푝 to denote the Eilenberg-MacLane spectrum.
• We let 풮 denote the ∞-category of spaces and Sp denote the ∞-category of spectra.
• We will call a space 푝-complete if it is Bousfield local with respect to F푝- homology, and we will call a spectrum 푝-complete if it is Bousfield local with respect to the Moore spectrum 푆/푝 [11, 12].
• We will call a space finite if it is homotopy equivalent to a finite complex. We will say a simply connected space is of finite type if its homotopy groups are finitely generated abelian groups in each degree.
21 22 Chapter 2
Equivariant Stable Homotopy Theory
In this chapter, we briefly review notions in equivariant stable homotopy theory that will play a critical role in this paper. We will use a variant of the framework of [4, §9], but we refer the reader to [24] for a classical treatment of the subject and [33] for another helpful account. The material in this chapter is not new, with the possible exception of Theorem 2.1.7, which the author thanks Jacob Lurie for suggesting. In §2.1, we will set notation by giving examples and motivation for the formal constructions related to global equivariance that follow in §2.2. Then in §2.3, we review the notion of Borel equivariant spectra and its relationship to Tate constructions.
2.1 Global equivariance
For a finite group 퐺, let Sp퐺 denote the ∞-category of genuine 퐺-spectra. This construction is contravariantly functorial in the group 퐺: for any group homomor- * phism 푓 : 퐻 → 퐾, one has a functor 푓 : Sp퐾 → Sp퐻 which can be thought of as follows:
* 1. For an injection 푓 : 퐻 ˓→ 퐺, 푓 : Sp퐺 → Sp퐻 can be thought of as restricting 퐺 to the subgroup 퐻, and so we will sometimes denote this functor by res퐻 .
* 2. For a surjection 푓 : 퐺 퐾, 푓 : Sp퐾 → Sp퐺 can be thought of as giving
23 a spectrum the trivial action on ker(푓). As such, we will sometimes denote 퐺 this functor by triv퐾 .
푓 ′ ′′ * 푓 In general, the functor 푓 can be computed by factoring 푓 as 퐻 −−− 퐺 ˓−→ 퐾 and setting 푓 * = 푓 ′* ∘ 푓 ′′*. One also has a covariant functoriality in the group 퐺 by multiplicative transfer:
3. For an injection 푔 : 퐻 ˓→ 퐺, one has a norm functor 푔⊗ : Sp퐻 → Sp퐺, due 퐺 to [18]. We will sometimes denote this functor by 푁퐻 .
4. For a surjection 푔 : 퐺 퐾, one has a geometric fixed point functor 푔⊗ : ker(푔) Sp퐺 → Sp퐾 , which we sometimes denote by Φ . We note that geometric fixed points commutes with all colimits.
Analogously to above, these are part of the same functoriality; one can func- torially associate a multiplicative transfer map 푔⊗ to any group homomorphism 푔 in a way that recovers the norm and geometric fixed point functors in the above cases.
′ ′′ 푔 푔 ′′ ′ Example 2.1.1. For the composite 푒 −→ 퐺 −→ 푒, we can check that 푔⊗ ∘ 푔⊗ = id = id⊗ because the geometric fixed points of the norm is the identity.
Variant 2.1.2. The ∞-categories Sp퐺 and the functors above depend only on the groupoid 퐵퐺 and not the group 퐺; in fact, one can make sense of all the above notions with groupoids in place of groups. Let Gpd denote the 2-category of groupoids 푋 such that 휋0(푋) and 휋1(푋) are finite. We will refer to objects of Gpd simply as groupoids. For 푋 ∈ Gpd, we let:
• Set푋 denote the category of 푋-sets.
• Fin푋 denote the category of finite 푋-sets.
• Sp푋 denote the ∞-category of genuine 푋-spectra ([4, §9]).
∐︀ 푋 ∏︀ Example 2.1.3. When 푋 = 퐵퐺푖, the ∞-category Sp is equivalent to Sp퐺푖 .
When 푋 = 퐵퐺 for a group 퐺, we may sometimes prefer to write Set퐺, Fin퐺, 퐵퐺 and Sp퐺. Note also that Sp does not denote the functor category Fun(퐵퐺, Sp).
24 For a map 푓 : 푋 → 푌 of groupoids, one has functors 푓 * : Sp푌 → Sp푋 and
푓⊗ : Sp푋 → Sp푌 which coincide with the similarly notated functors above when 푓 is induced by a group homomorphism. These two opposing functorialities interact as follows: let Span(Gpd) denote the ∞-category of spans of groupoids [8, §5]. Then, by [4, §9], there is a functor
Span(Gpd) → Cat∞
푓 푔 which sends a groupoid 푋 to the ∞-category Sp푋 and sends a span 푋 ←− 푀 −→ 푌 * 푋 푌 to the functor 푔⊗푓 : Sp → Sp . ∐︀ ∐︀ Example 2.1.4. When 푔 = 푔푖 : 푋푖 → 푌 is a map of groupoids, 푔⊗ is given by the tensor product of the (푔푖)⊗.
We use a variant of this framework which arises from modifying the ∞-category Span(Gpd):
Variant 2.1.5. In Definition 2.2.1, we construct a (2, 1)-category Glo+ which can be described informally as follows:
• The objects of Glo+ are groupoids 푋 ∈ Gpd.
• For two objects 푋, 푌 ∈ Glo+, the morphism groupoid Hom(푋, 푌 ) is the category of finite coverings 푀 → 푋 × 푌 and isomorphisms of coverings. We will think of a morphism as a span 푋 ← 푀 → 푌 .
• For two composable morphisms 푋 ← 푀 → 푌 and 푌 ← 푁 → 푍, the composite is the span 푋 ← 푇 → 푍 such that the map 푇 → 푋×푍 fits into the essentially unique factorization 푀 ×푌 푁 → 푇 → 푋 ×푍 of 푀 ×푌 푁 → 푋 ×푍 into a map with connected fibers followed by a finite cover.
We give a formal construction of Glo+ in Definition 2.2.1.
Remark 2.1.6. For groupoids 퐵퐻, 퐵퐺, 퐵퐾 ∈ Glo+, the morphism groupoid Glo+(퐵퐻, 퐵퐺) can be identified with the groupoid of (퐺, 퐻)-bisets and isomor- phisms, and the composition
Glo+(퐵퐺, 퐵퐾) × Glo+(퐵퐻, 퐵퐺) → Glo+(퐵퐻, 퐵퐾) sends a (퐾, 퐺)-biset 푇 and a (퐺, 퐻)-biset 푈 to the (퐾, 퐻)-biset 푇 ×퐺 푈.
25 Theorem 2.1.7. There is a functor
+ + Ψfun : Glo → Cat∞ sending a groupoid 푋 to the ∞-category Sp푋 of genuine 푋-spectra, and sending 푓 푔 * 푋 푌 a span 푋 ←− 푀 −→ 푌 to the functor 푔⊗푓 : Sp → Sp .
Example 2.1.8. The full subcategory of Glo+ spanned by discrete groupoids is equivalent to the (2, 1)-category Span(Fin) of spans of finite sets. The restricted functor Span(Fin) → Cat∞ sends a finite set 푇 to Fun(푇, Sp) and exhibits Sp as a symmetric monoidal ∞-category.
Example 2.1.9. Inside Glo+, we have the composite of morphisms
(퐵퐶푝 ← * → *) ∘ (* ← * → 퐵퐶푝) = (* ← 퐶푝 → *).
+ 퐶푝 Applying Ψfun, we recover the fact that the composite of the norm 푁푒 : Sp → 퐵퐶푝 퐶푝 퐵퐶푝 Sp and the restriction res푒 : Sp → Sp is the 푝-fold tensor product functor (−)⊗푝 : Sp → Sp.
Example 2.1.10. A more subtle example comes from composing the spans
(퐵퐶푝 ← 퐵퐶푝 → *) ∘ (* ← 퐵퐶푝 → 퐵퐶푝).
The composite as spans of groupoids would be (* ← 퐵퐶푝 → *). However, by the composition law of Glo+, this is replaced by (* ← * → *). This expresses the fact that for a spectrum 퐸, there is a canonical equivalence Φ퐶푝 triv퐶푝 퐸 ≃ 퐸. These equivalences Φ퐺 triv퐺 퐸 ≃ 퐸 are exactly the additional data captured by + Ψfun : Glo → Cat∞ but not by the corresponding functor from Span(Gpd) (see Remark 2.2.6).
Example 2.1.11. Let 퐻 ⊂ 퐺 be finite groups. Then the composite of spans
(퐵퐺 ← 퐵퐺 → *) ∘ (퐵퐻 ← 퐵퐻 → 퐵퐺) = (퐵퐻 ← 퐵퐻 → *)
퐻 퐺 퐺 expresses the fact that for 퐸 ∈ Sp , there is a canonical equivalence Φ 푁퐻 퐸 ≃ Φ퐻 퐸.
26 Example 2.1.12. Let 퐺 퐾 be a surjection of finite groups. Then the composite in Glo+
(퐵퐺 ← 퐵퐺 → *) ∘ (퐵퐾 ← 퐵퐺 → 퐵퐺) = (퐵퐾 ← 퐵퐾 → *)
퐾 퐺 퐺 expresses the fact that for 퐸 ∈ Sp , there is a canonical equivalence Φ triv퐾 퐸 ≃ Φ퐾 퐸.
2.2 Formal constructions around Glo+
We now give a formal definition of Glo+ and prove Theorem 2.1.7 (appearing here as Theorem 2.2.5).
+ Definition 2.2.1. Let the (2, 1)-category Glo ⊂ Cat1 be the subcategory of the (2, 1)-category of 1-categories whose objects are categories of the form Set푋 for 푋 ∈ Gpd and whose morphisms are functors which preserve limits and filtered colimits (equivalently, preserve compact objects).
Remark 2.2.2. By the adjoint functor theorem, the opposite category (Glo+)op can be described as the subcategory of Cat1 spanned by categories of the form Set푋 for 푋 ∈ Gpd and colimit preserving functors which send compact objects to compact objects. This acquires a symmetric monoidal structure under the tensor product, and so Glo+ also inherits a symmetric monoidal structure.
We will now unwind this definition of Glo+ to see that it admits the description given in Variant 2.1.5. The following lemma relates Definition 2.2.1 to Remark 2.1.6.
Lemma 2.2.3. Let 퐻 and 퐺 be finite groups. There is a fully faithful functor
op Fin퐻×퐺표푝 → Fun(Set퐻 , Set퐺) given by sending 푇 ∈ Fin퐻×퐺op to the functor 푋 ↦→ Hom퐻 (푇, 푋). The essential 푅,휔 image is the full subcategory Fun (Set퐻 , Set퐺) ⊂ Fun(Set퐻 , Set퐺) of functors which preserve limits and filtered colimits.
퐿 Proof. Let Fun (Set퐺, Set퐻 ) denote the full subcategory of left adjoint functors, which coincides with the full subcategory of colimit preserving functors by the
27 adjoint functor theorem. Since Set퐺 is freely generated under colimits by the free transitive 퐺-set, we have the equivalences
퐿 op Fun (Set퐺, Set퐻 ) ≃ Fun(퐵퐺 , Set퐻 ) 표푝 ≃ Fun(퐵퐻 × 퐵퐺 , Set) ≃ Set퐻×퐺op .
By taking adjoints, we obtain a fully faithful embedding
푅,휔 퐿 op op Fun (Set퐻 , Set퐺) ⊂ Fun (Set퐺, Set퐻 ) ≃ Set퐻×퐺op with essential image exactly the finite 퐺-퐻-bisets.
We now relate Glo+ to the ∞-category Span(Gpd). Recall that the latter has the structure of a symmetric monoidal ∞-category under the Cartesian product of groupoids [9, §7].
Proposition 2.2.4. There is a symmetric monoidal functor
휋 : Span(Gpd) → Glo+
푓 푔 which sends a groupoid 푋 to the category Set푋 and sends a span (푋 ←− 푀 −→ 푌 ) * 푋 푌 * to the functor 푔×푓 : Set → Set , where 푔× denotes the right adjoint to 푔 .
Proof. The (Set-valued) Yoneda embedding gives a functor Gpdop → Glo+ which takes products of groupoids to tensor products of presentable categories. The restriction functors admit right adjoints satisfying the necessary Beck-Chevalley conditions, so by [9, Construction 7.6], we obtain the desired symmetric monoidal functor 휋 : Span(Gpd) → Glo+.
Convention. It follows from Lemma 2.2.3 that for 푋, 푌 ∈ Glo+, the groupoid Glo+(푋, 푌 ) can be identified with the groupoid of finite coverings of 푋 ×푌 . Under this identification, the functor 휋 of Proposition 2.2.4 sends a span 푀˜ → 푋 × 푌 to the span 푀 → 푋 ×푌 determined by the canonical factorization 푀˜ → 푀 → 푋 ×푌 of 푀˜ → 푋 × 푌 into a map with connected fiber followed by a finite cover. This justifies the description given in Variant 2.1.5. We will, therefore, continue to think of the 2-category Glo+ as groupoids with certain spans between them unless otherwise specified.
28 We now prove the main result of the chapter:
Theorem 2.2.5. There is a coCartesian fibration Ψ+ : Glo+Sp → Glo+ such that 푓 푔 the fiber over 푋 ∈ Glo+ is Sp푋 , and for any span 휎 = (푋 ←− 푀 −→ 푌 ), the 푋 푌 * associated functor 휎* : Sp → Sp is given by 푔⊗ ∘ 푓 .
+ + Proof. There is a tautological functor Glo → Cat1 sending an object of Glo , thought of as a category, to its subcategory of compact objects. Thinking of Glo+ in terms of groupoids, this sends, for instance, 퐵퐺 to Fin퐺. The procedure of [4, §9.2] then applies verbatim.
Remark 2.2.6. Theorem 2.2.5 is a strengthening of the constructions in [4, Section 9.2], which produces the restriction of the fibration Ψ+ along the functor 휋 : Span(Gpd) → Glo+ of Proposition 2.2.4. The extension of this fibration to Glo+ encodes exactly the additional fact that geometric fixed points is inverse to giving trivial action, as explained in Example 2.1.10. This additional data turns out to power the construction of the Frobenius action in §3.
We will also be interested in a subcategory of Glo+ which restricts the types of multiplicative transfers that are allowed:
Definition 2.2.7. Let Glo ⊂ Glo+ be the wide subcategory where the morphisms 푓 푔 are spans (푋 ←− 푀 −→ 푌 ) which have the property that the forward map 푔 has discrete fibers. Note that Glo is closed under the monoidal structure on Glo+ and therefore inherits a symmetric monoidal structure. We shall denote the restriction of Ψ+ along the inclusion by Ψ : GloSp → Glo .
Remark 2.2.8. Note that Glo is also a subcategory of Span(Gpd). Thus, for the purposes of defining Glo, one can simply start with the ∞-category Span(Gpd). However, we will see that in order to get the Frobenius action, it will be important that Glo is embedded inside Glo+ rather than just Span(Gpd).
2.3 Borel equivariant spectra and proper Tate con- structions
Let 푋 ∈ Gpd be a groupoid. Then for every point in 푋, one obtains a restric- tion functor Sp푋 → Sp. These assemble to a functor Sp푋 → Fun(푋, Sp) which
29 we will think of as taking a genuine equivariant 푋-spectrum to its underlying spectrum (with group action). This functor admits a fully faithful right adjoint 푋 푗푋 : Fun(푋, Sp) ˓→ Sp which we think of as cofreely promoting a non-genuine spectrum to a genuine spectrum. The essential image of this embedding is known 푋 푋 as the full subcategory SpBor ⊂ Sp of Borel 푋-spectra. We will also abusively refer to objects of Fun(푋, Sp) as Borel 푋-spectra. We write 훽푋 , or simply 훽, for the composite 푗 Sp푋 → Fun(푋, Sp) −→푋 Sp푋 , so every 퐸 ∈ Sp푋 admits a natural Borelification map 퐸 → 훽퐸.
Example 2.3.1. The data of a genuine 퐶푝-spectrum 퐸 can be presented as a triple (퐸0, 퐸1, 푓) where 퐸0 ∈ Fun(퐵퐶푝, Sp) is the underlying spectrum, 퐸1 ∈ Sp 퐶푝 푡퐶푝 is thought of as Φ 퐸, and 푓 : 퐸1 → 퐸0 is a map of spectra. From this data, one recovers the genuine fixed points of 퐸 via the homotopy pullback square
퐶푝 퐸 퐸1
푓
ℎ퐶푝 푡퐶푝 퐸0 퐸0 .
The genuine 퐶푝-spectrum 퐸 is Borel if and only if the natural map 퐸 → 훽퐸 is an equivalence of genuine 퐶푝-spectra. This map induces an equivalence on underlying spectra, so it is a genuine equivalence if and only if it is an equivalence after applying Φ퐶푝 . Since Φ퐶푝 훽퐸 = 퐸푡퐶푝 , this is equivalent to the condition that 푡퐶푝 the specified map 푓 : 퐸1 → 퐸0 is an equivalence.
We will need a variant of the Tate construction.
Definition 2.3.2 ([2], Definition 2.7). For a finite group 퐺, let
휏퐺 (−) : Sp퐺 → Sp denote the functor given by the formula 퐸휏퐺 = Φ퐺(훽푋). We will refer to (−)휏퐺 as the proper Tate construction for 퐺. When 퐺 = 퐶푝, this agrees with the ordinary Tate construction.
Remark 2.3.3. The proper Tate construction is a lax monoidal functor because both Φ퐺 and 훽 are.
30 Remark 2.3.4. Just as the Tate construction can be thought of as universally killing 퐺-spectra of the form 퐺+ ⊗퐸, the proper Tate construction can be thought of as universally killing 퐺-spectra induced from any proper subgroup (i.e., of the form (퐺/퐻)+ ⊗ 퐸 for 퐻 ( 퐺). This description is made precise in [2, Remark 2.16] using Verdier quotients of stable ∞-categories.
Remark 2.3.5. It is clear from the definition that the proper Tate construction depends only on the underlying spectrum with 퐺-action. Unwinding the definition of geometric fixed points, one can give an alternate description of the proper Tate construction analogous to the description of the usual Tate construction as the cofiber of the additive norm map. Let 퐺 be a finite group, let 풪퐺 denote the − category of finite transitive 퐺-sets, and let 풪퐺 ⊂ 풪퐺 denote the full subcategory spanned by 퐺-sets with nontrivial action. Then, a spectrum 푋 with 퐺-action ℎ퐻 determines a functor 풯 : 풪퐺 → Sp which sends 퐺/퐻 to 푋 and sends maps of 퐺-sets to the corresponding additive transfer maps. The spectrum 푋휏퐺 is the total cofiber of this diagram; i.e., it fits into the cofiber sequence
colim 풯 → 푋ℎ퐺 → 푋휏퐺. − 풪퐺
Example 2.3.6 (Tate diagonals). Let 퐺 be a finite group. There is a canonical 퐺 퐺 lax monoidal natural transformation 푁 (−) → 훽푁 (−) of functors Sp → Sp퐺. Applying geometric fixed points, this yields a natural transformation
∆퐺 : id ≃ Φ퐺푁 퐺(−) → Φ퐺훽푁 퐺(−) ≃ ((−)⊗퐺)휏퐺 which we call the Tate diagonal for 퐺. This has been considered previously in [16, 34, 22, 2].
31 32 Chapter 3
The Integral Frobenius Action
In this chapter, we prove Theorem A♮ by building what we call the integral Frobe- nius action. This is an oplax action of the symmetric monoidal category 풬 on the ∞-category CAlg of E∞-rings which captures the functoriality of the E∞- Frobenius.
In §3.1, we begin by recalling the E∞-Frobenius map from [34]. We then gen- eralize this construction in §3.2 to produce certain “generalized Frobenius maps” associated to inclusions 퐻 ⊂ 퐺 of finite groups and use them to give a precise statement of Theorem A♮ (Theorem 3.2.7). The remainder of the chapter is ded- icated to proving this theorem. In §3.3, we construct the ∞-category CAlgGlo of global algebras, which are roughly E∞-rings 퐴 together with “genuine equivariant multiplications” that we think of as specifying lifts of the generalized Frobenius maps to endomorphisms of 퐴. We show that these Frobenius lifts assemble into an action of 풬 on CAlgGlo (Theorem 3.3.6). It then remains to descend this action to an action on the ∞-category CAlg. We accomplish this in §3.4 by introducing a structure mediating between E∞-rings and global algebras: we call these Borel Glo global algebras and denote the ∞-category of them by CAlgBor. These Borel global algebras are, on the one hand, close enough to global algebras that one obtains Glo an oplax action of 풬 on CAlgBor. On the other hand, the main result of §3.4 is Glo that CAlgBor is actually equivalent to the ∞-category of E∞-rings. Together, these facts imply Theorem A♮.
33 3.1 The E∞-Frobenius
Example 3.1.1. Let 푅 be a discrete commutative ring and let 푝 be a prime. Then one has two natural ring maps 푅 → 푅/푝: the quotient map and the 푝th power map.
There are analogous maps for an E∞-ring 퐴, which we now recall (following [34]).
푡퐶푝 Definition 3.1.2. Let can퐴 : 퐴 → 퐴 denote the composite
퐴 → 퐴ℎ퐶푝 → 퐴푡퐶푝 where the first map is the restriction induced by the map 퐵퐶푝 → * and the second map is the canonical projection to the Tate construction. We refer to can퐴 as the canonical map and remark that can퐴 makes sense for any spectrum 퐴 but is a ring map if 퐴 is a ring.
푡퐶푝 Definition 3.1.3. Let 휙퐴 : 퐴 → 퐴 denote the composite
Δ ⊗푝 푡퐶푝 푡퐶푝 휙퐴 : 퐴 −→ (퐴 ) → 퐴 of the Tate diagonal followed by multiplication. This is a ring map because ∆ and (−)푡퐶푝 are lax monoidal (cf. Remark 2.3.3 and Example 2.3.6). We will refer to this map throughout as the E∞-Frobenius, or simply the Frobenius.
Example 3.1.4. When 퐴 = 푅 is a discrete commutative ring, the canonical map 푡퐶푝 푡퐶푝 푅 → 푅 on 휋0 exhibits 휋0(푅 ) as the quotient 푅/푝. Under this equivalence, the Frobenius on 휋0 is given by the 푝th power map 푅 → 푅/푝.
However, the E∞-Frobenius for a discrete ring is not the map induced by the ordinary Frobenius. Instead, one sees the Steenrod operations in higher homotopy groups.
푡퐶2 Example 3.1.5 ([34], Theorem IV.1.15). When 퐴 = F2, we have that 휋*퐴 ≃ F2((푡)) where |푡| = −1. The Frobenius
푡퐶2 휙F2 : F2 → (F2)
34 is given, as a map of spectra, by the product over 푖 ≥ 0 of
푖 푖 Sq : F2 → Σ F2 .
More generally, if 퐴 is an E∞-algebra over F2, we have that the Frobenius is given on homotopy groups by the map
휋*(퐴) → 휋*(퐴)((푡)) ∑︁ 푥 ↦→ Sq푖(푥)푡−푖. 푖∈Z
In general, the Frobenius can be thought of as a “total degree 푝 stable power operation.” Another basic example that will be important to us is the following:
Example 3.1.6. When 퐴 = 푆, the sphere spectrum, the Frobenius map and the canonical map are both the unique ring map 푆 → 푆푡퐶푝 , which exhibits 푆푡퐶푝 as the 푝-completion of 푆 by Lin’s theorem [25].
3.2 Generalized Frobenius and canonical maps
As explained in Remark 1.2.8, Theorem A heuristically says that the Frobenius “commutes with itself.” To prove this, one has to understand all ways of composing the Frobenius with itself. We accomplish this by creating a Frobenius composite 퐺 휏퐺 퐺 휏퐻 휏퐺 휙 : 퐴 → 퐴 for every finite group 퐺 (and, in fact, a map 휙퐻 : 퐴 → 퐴 for any inclusion 퐻 ⊂ 퐺 of finite groups), and then describing how these interact. Roughly, if |퐺| = 푝푘, one can think of 휙퐺 as an approximation to the 푘-fold 푡퐶푝 composite of the E∞-Frobenius 휙 : 퐴 → 퐴 .
Construction 3.2.1 (Generalized Frobenius maps). Let 퐻 ⊂ 퐺 be an inclusion of finite groups and let 퐴 ∈ CAlg be an E∞-ring spectrum. Then, there is a natural map 퐴⊗퐺/퐻 → 퐴 of spectra with 퐺-action. We regard 퐴⊗퐺/퐻 as the underlying 퐺 퐻 spectrum of the genuine 퐺-spectrum 푁퐻 (훽퐻 triv 퐴). By the universal property of Borel 퐺-spectra, one obtains a natural map
퐺 퐻 퐺 푁퐻 (훽퐻 triv 퐴) → 훽퐺 triv 퐴.
Applying Φ퐺 and using the equivalence of Example 2.1.11, we obtain a generalized
35 Frobenius map
퐺 휏퐻 퐻 퐻 퐺 퐺 퐻 퐺 퐺 휏퐺 휙퐻 : 퐴 ≃ Φ (훽퐻 triv 퐴) ≃ Φ 푁퐻 (훽퐻 triv 퐴) → Φ (훽퐺 triv 퐴) ≃ 퐴 .
Remark 3.2.2. In the special case 퐻 = *, the generalized Frobenius map 휙퐺 is given by the composite 퐺 퐴 −−→Δ (퐴⊗퐺)휏퐺 → 퐴휏퐺 of the Tate diagonal (cf. Example 2.3.6) with multiplication. In particular, taking 퐺 = 퐶푝, we recover the E∞-Frobenius. There are corresponding generalizations of the canonical maps.
Construction 3.2.3 (Generalized canonical maps). Let 퐺 퐾 be a surjection of finite groups and let 퐴 ∈ CAlg be an E∞-ring spectrum. Then one has a genuine 퐺-equivariant map 퐺 퐾 퐺 triv퐾 (훽퐾 triv 퐴) → 훽퐺 triv 퐴 by the universal property of Borel 퐺-spectra. Applying Φ퐺 and using the equiva- lence of Example 2.1.12, we obtain a generalized canonical map
퐺 휏퐾 휏퐺 can퐾 : 퐴 → 퐴 .
Remark 3.2.4. In the special case 퐾 = *, the generalized canonical map can퐺 is given by the composite 퐴 → 퐴ℎ퐺 → 퐴휏퐺.
In particular, when 퐺 = 퐶푝, we recover the canonical map of Definition 3.1.2.
We see that while the generalized Frobenius maps are (covariantly) functo- rial for injections of groups, the generalized canonical maps are contravariantly functorial for surjections of groups. The interaction between these opposing func- torialities is captured by a variant of Quillen’s 푄-construction:
Definition 3.2.5. Let 풬 be the symmetric monoidal 1-category defined as follows:
• The objects of 풬 are finite groups.
• For finite groups 퐺, 퐻, the set Hom풬(퐻, 퐺) is the set of isomorphism classes of spans of finite groups (퐻 퐾 ˓→ 퐺) where the left morphism is a surjection and the right morphism is an injection.
36 • Composition of morphisms is the usual composition of spans.
• The symmetric monoidal structure is by Cartesian product of groups.
Remark 3.2.6. Strictly speaking, the category of finite groups is not an exact category, and so 풬 is not an example of Quillen’s 푄-construction. However, the full subcategory of 풬 spanned by the finite abelian groups is Quillen’s Q-construction [38] on the exact category of abelian groups.
We may now state Theorem A♮ more precisely:
Theorem 3.2.7 (Integral Frobenius action). There is an oplax monoidal functor
Θ: 풬 → Fun(CAlg, CAlg) with the following properties:
• The object 퐺 ∈ 풬 acts by the functor (−)휏퐺 : CAlg → CAlg.
• A left morphism (퐾 퐺 → 퐺) in 풬 is sent to the natural transformation 퐺 휏퐾 휏퐺 can퐾 :(−) → (−) .
• A right morphism (퐻 ← 퐻 ˓→ 퐺) in 풬 is sent to the natural transformation 퐺 휏퐻 휏퐺 휙퐻 :(−) → (−) .
Remark 3.2.8. The oplax structure corresponds to a natural map 퐴휏퐺×퐻 → 휏퐺 휏퐻 (퐴 ) for any E∞-ring 퐴 and finite groups 퐺, 퐻. This is most directly seen by the universal property of the proper Tate construction (cf. Remark 2.3.4), but one may also give a description in the spirit of Constructions 3.2.1 and 3.2.3 as 퐻 퐺×퐻 follows. We observe that the underlying Borel 퐺-spectrum of Φ 훽퐺×퐻 triv 퐴 퐺 퐻 휏퐻 and triv Φ 훽퐻 퐴 are both 퐴 with the trivial 퐺 action. This yields a canonical genuine 퐺-equivariant map
퐻 퐺×퐻 퐺 퐻 Φ 훽퐺×퐻 triv 퐴 → 훽퐺 triv Φ 훽퐻 퐴, from which we extract the desired map as the composite
휏퐺×퐻 퐺 퐻 퐺×퐻 퐺 퐺 퐻 휏퐻 휏퐺 퐴 ≃ Φ Φ 훽퐺×퐻 triv 퐴 → Φ 훽퐺 triv Φ 훽퐻 퐴 ≃ (퐴 ) .
37 Remark 3.2.9. Recall from Example 3.1.5 that the Frobenius map for an F2- algebra 퐴 captures the total Steenrod operation. Unwinding the functoriality of Theorem 3.2.7 on the full subcategory of 풬 spanned by the groups *, 퐶2 and 퐶2 × 퐶2, one recovers the proof of the Adem relations given in [13] and [46], with the additional feature that one does not need to explicitly mention the groups Σ4 or Σ2 ≀ Σ2. More generally, thinking of the Frobenius as a “total stable power operation,” this theorem can be seen as expressing the higher relations that occur when one composes stable power operations.
The remainder of the chapter is dedicated to proving Theorem 3.2.7.
3.3 Global algebras
Let 퐺 be a finite group. Then any E∞-ring spectrum 퐴 comes equipped with a ⊗퐺 multiplication map (퐴 )ℎ퐺 → 퐴. However, one could ask for a stronger notion of multiplication.
Question. Does the multiplication on 퐴 canonically lift to a genuine 퐺-equivariant multiplication map 푁 퐺퐴 → triv퐺 퐴?
In general, the answer is no – one needs to specify more data:
Example 3.3.1. Consider the case 퐺 = 퐶푝. Recall that the data of a genuine 퐶푝 퐶푝-spectrum 푋 is the data of a spectrum 푋 with 퐶푝-action, a spectrum Φ 푋, and 퐶푝 푡퐶푝 퐶푝 퐶푝 a map Φ 푋 → 푋 . To specify a genuine 퐶푝-equivariant map 푁 퐴 → triv 퐴 lifting the E∞ multiplication on 퐴 is to fill in the dotted arrow in the following diagram:
Φ퐶푝 푁 퐶푝 퐴 퐴 퐴 Φ퐶푝 triv퐶푝 퐴
Δ can
(퐴⊗푝)푡퐶푝 퐴푡퐶푝
where the bottom arrow is induced by the E∞ structure. In other words, one 푡퐶푝 needs a lift of the E∞-Frobenius map 퐴 → 퐴 to an endomorphism of 퐴. In general, a genuine 퐺-equivariant multiplication in the above sense gives 퐴 the
38 dotted “Frobenius lift” in the following diagram:
퐴 퐴
can퐺 휙퐺 퐴휏퐺.
We now define the notion ofa global algebra, which is roughly an E∞-ring spectrum together with coherent choices of genuine 퐺-equivariant multiplication for all 퐺.
Definition 3.3.2. A global algebra is a section of the fibration Ψ which is co- Cartesian over the left morphisms. We let CAlgGlo denote the ∞-category of global algebras.
Remark 3.3.3. Concretely, such a section 퐴 : Glo → GloSp is an assignment to each groupoid 푋 ∈ Glo a genuine equivariant spectrum 퐴(푋) ∈ Sp푋 together with certain structure maps. The condition that the section is coCartesian over the left morphisms implies in particular that 퐴(퐵퐺) ≃ triv퐺 퐴(*) (by considering the span (* ← 퐵퐺 → 퐵퐺)), so that the value of the section on a point determines the value of the section on any other groupoid. The functoriality in the span (* ← * → 퐵퐺) encodes the genuine multiplication maps
퐺 퐺 푁 퐴(*) ≃ (* ← * → 퐵퐺)*퐴(*) → 퐴(퐵퐺) ≃ triv 퐴.
Remark 3.3.4. The restriction of a global algebra 퐴 along the inclusion Span(Fin) → Glo of Example 2.1.8 gives 퐴(*) the structure of an E∞-ring spectrum by [14, Sec- tion 5]. We will think of 퐴(*) as the underlying E∞-ring of 퐴.
By Example 3.3.1, we can think of a global algebra as an E∞-ring together with “Frobenius lifts." The functoriality of these Frobenius lifts is explained by the following observation:
Observation 3.3.5. Let 퐴 ∈ CAlgGlo and 퐺 be a finite group. Then one can form another global algebra 퐴퐺, which we think of as 퐴 twisted by 퐺, via the formula
퐺 퐴 (푋) = 휎*퐴(푋 × 퐵퐺),
proj where 휎 denotes the morphism (푋 × 퐵퐺 ← 푋 × 퐵퐺 −−−→1 푋) in Glo+. More
39 concretely, we have for any finite group 퐻 that
퐴퐺(퐵퐻) = Φ퐺퐴(퐵퐻 × 퐵퐺).
Moreover, these twists are functorial in certain maps in the finite group 퐺. 1. For an injection 퐻 → 퐺, we have a natural map 퐴퐻 → 퐴퐺 given by:
퐻 퐴 (푋) = (푋 × 퐵퐻 ← 푋 × 퐵퐻 → 푋)*퐴(푋 × 퐵퐻)
≃ (푋 × 퐵퐺 ← 푋 × 퐵퐺 → 푋)*(푋 × 퐵퐻 ← 푋 × 퐵퐻 → 푋 × 퐵퐺)*퐴(푋 × 퐵퐻) 퐺 → (푋 × 퐵퐺 ← 푋 × 퐵퐺 → 푋)*퐴(푋 × 퐵퐺) = 퐴 (푋)
where we have used that 퐴 is a section over Glo and that 푋 × 퐵퐻 → 푋 × 퐵퐺 has discrete fibers.
2. For a surjection 퐺 → 퐾, we have a natural map 퐴퐾 → 퐴퐺 given by:
퐾 퐴 (푋) = (푋 × 퐵퐾 ← 푋 × 퐵퐾 → 푋)*퐴(푋 × 퐵퐾)
≃ (푋 × 퐵퐺 ← 푋 × 퐵퐺 → 푋)*(푋 × 퐵퐾 ← 푋 × 퐵퐺 → 푋 × 퐵퐺)*퐴(푋 × 퐵퐾) 퐺 → (푋 × 퐵퐺 ← 푋 × 퐵퐺 → 푋)*퐴(푋 × 퐵퐺) = 퐴 (푋)
where we note that we have used critically the definition of the composition law of Glo+ and the fact that 푋 × 퐵퐺 → 푋 × 퐵퐾 has connected fibers. The main result of this section is that these two opposing functorialities as- semble into an action of 풬 on global algebras. In fact, the twists make sense for any section of Ψ – not just global algebras; we therefore state our theorem more generally: Theorem 3.3.6. There is a monoidal functor
풬 → Fun(sect(Ψ), sect(Ψ)) sending a group 퐺 to the functor 퐴 ↦→ 퐴퐺 and such that maps in 풬 are sent to the corresponding maps identified in Observation 3.3.5. This action fixes thefull subcategory of global algebras, and thus determines an action of 풬 on CAlgGlo. Proof. By Proposition A.0.10 applied to the coCartesian fibration 푞+ = Ψ+ : Glo+Sp → Glo+ and the subcategory 푖 = 휄 : Glo → Glo+, we learn that the
40 ∞-category sect(Ψ) admits a natural right action of the monoidal ∞-category + Fun(Glo, Glo) ×Fun(Glo,Glo+) Fun(Glo, Glo )/휄. The proof is completed by the fol- lowing proposition, which says that this action can be restricted to an action of 풬, which admits the description given in Observation 3.3.5.
Proposition 3.3.7. There is a monoidal functor
+ 풬 → Fun(Glo, Glo) ×Fun(Glo,Glo+) Fun(Glo, Glo )/휄 sending a group 퐺 to the functor (−) × 퐵퐺 : Glo → Glo given by multiplication by 퐵퐺 together with the natural transformation 휄(−) × 퐵퐺 → 휄(−) of functors Glo → Glo+ given on 푋 ∈ Glo by the span (푋 × 퐵퐺 ← 푋 × 퐵퐺 → 푋).
Proof. The symmetric monoidal inclusion 휄 : Glo → Glo+ endows Glo+ with the structure of a module over the symmetric monoidal ∞-category Glo. The ∞-category ModGlo(Cat∞) of Glo-module ∞-categories is naturally tensored over Cat∞ via the Cartesian product and the forgetful functor ModGlo(Cat∞) → Cat∞ respects this tensoring.
Recall the Cartesian fibration ℳ → Cat∞ of Construction A.0.4. We make the following variant: let 풩 → ModGlo(Cat∞) denote the Cartesian fibration classified op + op by the functor ModGlo(Cat∞) → Cat∞ which sends 풞 to FunGlo(풞, Glo ) , where the subscript denotes the ∞-category of Glo-module maps. The forgetful natural transformation + op + op FunGlo(풞, Glo ) → Fun(풞, Glo ) yields a functor 푈 : 풩 → ℳ. By construction, the functor 푈 respects the tensoring + over Cat∞. Moreover, the inclusion 휄 : Glo → Glo is a map of Glo-modules, so (Glo, 휄) ∈ ℳ admits a natural lift (Glo, 휄) ∈ 풩 . Consequently, there is a monoidal functor
+ End풩 (Glo, 휄) → Endℳ(Glo, 휄) = Fun(Glo, Glo) ×Fun(Glo,Glo+) Fun(Glo, Glo )/휄.
It therefore suffices to produce an appropriate monoidal functor 풬 → End풩 (Glo, 휄). By an analogous argument to Lemma A.0.5, we may identify the underlying ∞-
41 category
Glo Glo + End풩 (Glo, 휄) ≃ Fun (Glo, Glo) ×FunGlo(Glo,Glo+) Fun (Glo, Glo )/휄 + ≃ Glo ×Glo+ Glo /* .
Let us analyze the natural projection
+ 푝 : End풩 (Glo, 휄) ≃ Glo ×Glo+ Glo /* → Glo, which has the structure of a monoidal functor by construction. Consider the following subcategories:
+ 1. Let 풟 ⊂ Glo ×Glo+ Glo /* be the full subcategory spanned by spans of the form (푋 = 푋 → *) such that 푋 is connected. For ease of notation, we may refer to this object of 풟 simply as 푋.
2. Let 풬˜ ⊂ Glo be the subcategory spanned by the connected groupoids and 푓 푔 morphisms of the form (푋 ←− 푀 −→ 푌 ) where 푓 has connected fibers and 푔 has discrete fibers. In other words, the spans take the form 퐵퐻 ← 퐵퐺 → 퐵퐾 where 퐺 → 퐻 is surjective and 퐺 → 퐾 is injective.
It is immediate that 풟 and 풬˜ are monoidal subcategories, and that the projection 푝 restricts to a monoidal functor 푝 : 풟 → 풬˜.
Lemma 3.3.8. The restricted functor 푝 : 풟 → 풬˜ is an equivalence of monoidal ∞-categories.
Proof. Essential surjectivity is obvious so it suffices to show that 푝 : 풟 → 풬˜ is fully faithful. For this, consider two objects of 풟 corresponding to the groupoids 퐵퐻 and 퐵퐾 together with the natural maps 푝퐻 = (퐵퐻 = 퐵퐻 → *) and 푝퐾 = (퐵퐾 = 퐵퐾 → *) in Glo+. We would like to show that the natural map
Hom (퐵퐻, 퐵퐾) × Hom + (퐵퐻, 퐵퐾) → Hom ˜(퐵퐻, 퐵퐾) Glo HomGlo+ (퐵퐻,퐵퐾) Glo /* 풬 is an equivalence of groupoids. We first observe that any map 퐵퐻 → 퐵퐾 in Glo which is compatible with the natural maps 퐵퐻 → * and 퐵퐺 → * in Glo+ is automatically in the subcategory 풬˜ ⊂ Glo. Thus, it suffices to show the natural
42 map
Hom ˜(퐵퐻, 퐵퐾) × Hom + (퐵퐻, 퐵퐾) → Hom ˜(퐵퐻, 퐵퐾) 풬 HomGlo+ (퐵퐻,퐵퐾) Glo /* 풬 (3.1) is an equivalence of groupoids. But we have that
Hom + (퐵퐻, 퐵퐾) ≃ Hom + (퐵퐻, 퐵퐾) × {푝 }. Glo /* Glo HomGlo+ (퐵퐻,*) 퐻
The lemma then follows by noting that for any 휎 ∈ Hom풬˜(퐵퐻, 퐵퐾), we have op ≃ 푝퐻 = 푝퐾 ∘휎 so (3.1) is essentially surjective, and 푝퐻 ∈ HomGlo+ (퐵퐻, *) ≃ (Fin퐻 ) has no automorphisms (since it corresponds to the singleton as an 퐻-set) so (3.1) is fully faithful.
To complete the proof of Proposition 3.3.7 and with it, Theorem 3.3.6, we consider the functor 풬 → Glo given by 퐺 ↦→ 퐵퐺. This functor is monoidal because Glo is a monoidal subcategory of Span(Gpd) (Remark 2.2.8) and 풬 is a monoidal subcategory of spans of groups. Since this monoidal functor factors through the subcategory 풬˜, we may restrict to obtain the desired monoidal functor
풬 → Fun(CAlgGlo, CAlgGlo).
3.4 Borel global algebras
In this section, we complete the proof of Theorem 3.2.7, which concerns producing an oplax action of 풬 on CAlg. We will deduce this from Theorem 3.3.6 by studying the difference between E∞-rings and global algebras. Let 퐴 ∈ CAlg be an E∞-ring spectrum. We have seen that 퐴 does not canon- ically determine a global algebra because 퐴 does not come with a genuine 퐶푝- equivariant map 푁 퐶푝 퐴 → triv퐶푝 퐴. However, 퐴 does have a weaker version of this 퐶푝 퐶푝 structure: the data of a genuine 퐶푝-equivariant map 푁 퐴 → 훽퐶푝 triv 퐴, because ⊗푝 this is the same data as a map 퐴 → 퐴 of spectra with 퐶푝-action. The notion of a Borel global algebra is the analog of a global algebra where one demands this weaker structure.
43 Glo After constructing the ∞-category CAlgBor of Borel global algebras, we show Glo that the action of Theorem 3.3.6 descends to an oplax action of 풬 on CAlgBor (Proposition 3.4.7). On the other hand, we prove that sending a Borel global al- Glo gebra to its underlying E∞-ring induces an equivalence of ∞-categories CAlgBor ≃ CAlg (Theorem 3.4.8). Together, these two facts imply Theorem 3.2.7.
Definition 3.4.1. For each 푋 ∈ Glo, recall that there is a full subcategory 푗푋 : 푋 푋 SpBor ˓→ Sp of Borel 푋-spectra. Let GloSpBor ⊂ GloSp be the full simplicial 푋 subset on the pairs (퐸, 푋) ∈ GloSp with the property that 퐸 ∈ SpBor. This yields a diagram
푗 GloSpBor GloSp
ΨBor Ψ (3.2) Glo .
Proposition 3.4.2. The map ΨBor is a coCartesian fibration.
Proof. The map ΨBor is an inner fibration because it is a full simplicial subset of an inner fibration. 푋 Let 휎 = (푋 ← 푆 → 푌 ) be a morphism in Glo and 퐸 ∈ SpBor. Then, 휎 has a coCartesian lift given by the arrow (퐸, 푋) → (훽푌 휎*퐸, 푌 ) in GloSpBor induced by the natural map 휎*퐸 → 훽휎*퐸. It follows that ΨBor is a locally coCartesian fibration. But these locally coCartesian arrows are closed under composition: for any 휏 = (푌 ← 푇 → 푍) in Glo, we have that the natural transformation 푌 푍 푋 훽푍 휏* → 훽푍 휏*훽푌 of functors Sp → Sp is an equivalence, so for any 퐸 ∈ SpBor, the natural map
훽푍 (휏 ∘ 휎)*퐸 → 훽푍 휏*훽푌 휎*퐸 is an equivalence (note that it is important that we are working with Ψ and not + + all of Ψ because this may not be true if 휏 is in Glo but not Glo.). Thus, ΨBor is a coCartesian fibration by [29, Proposition 2.4.2.8].
Warning 3.4.3. The inclusion GloSpBor ⊂ GloSp does not send coCartesian ar- rows to coCartesian arrows because the natural map 휎*퐸 → 훽휎*퐸 need not be an equivalence.
Lemma 3.4.4. The inclusion 푗 : GloSpBor → GloSp of diagram (3.2) admits a left adjoint relative to Glo in the sense of [28, Definition 7.3.2.2].
44 Proof. We apply [28, Proposition 7.3.2.11]. The functors Ψ and ΨBor are coCarte- sian fibrations by Theorem 2.2.5 and Proposition 3.4.2, and therefore theyare locally coCartesian categorical fibrations. For condition (1), we need only recall 푋 푋 that for each 푋 ∈ Glo, the inclusion 푗푋 : SpBor → Sp admits a left adjoint
푋 푋 훽푋 : Sp ↽−⇀− SpBor : 푗푋 .
Condition (2) amounts to the fact that the genuine equivariant norm and restric- tion maps coincide with the corresponding Borel norm and restriction maps on the underlying Borel equivariant spectrum.
Corollary 3.4.4.1. There is an adjunction
푠 푠 훽 : sect(Ψ) ↽−⇀− sect(ΨBor): 푗 at the level of sections which restricts to the adjunction
푋 푋 훽푋 : Sp ↽−⇀− SpBor : 푗푋 over each 푋 ∈ Glo .
We now make the analog of Definition 3.3.2 in this setting.
Definition 3.4.5. A Borel global algebra is a section of the fibration ΨBor which Glo is coCartesian over the left morphisms. We let CAlgBor denote the ∞-category of Borel global algebras.
Remark 3.4.6. A Borel global algebra can be thought of as an E∞ algebra 퐴(퐵퐺) ∈ Fun(퐵퐺, Sp) for every groupoid 퐵퐺 together with structure maps cor- responding to the various maps in Glo. As in the case of global algebras, 퐴(*) acquires the structure of an E∞-ring spectrum (cf. Remark 3.3.4) and determines the value of 퐴 at any other 푋 ∈ Glo . The right morphisms then encode certain multiplication maps; for instance the span (* ← * → 퐵퐶푝) encodes a map
훽푁 퐶푝 퐴(*) → 훽 triv퐶푝 퐴(*) in Fun(퐵퐶푝, 푆푝). We saw at the beginning of the section that this map is already part of the E∞-structure on 퐴(*). In fact, we will show in Theorem 3.4.8 that the
45 additional structure of a Borel global algebra is specified uniquely by 퐴(*) as an E∞-ring.
Theorem 3.2.7 follows immediately from the following two statements about Glo CAlgBor:
Proposition 3.4.7. There is an oplax monoidal functor
Glo Glo 풬 → Fun(CAlgBor, CAlgBor) which admits the description of Theorem 3.2.7 on underlying E∞-rings.
Proof. By Theorem 3.3.6 and Lemma A.0.1 applied to the adjunction of Corollary 3.4.4.1, we obtain an oplax right action of 풬 on sect(ΨBor); since 풬 is a symmetric monoidal category, left and right actions are the same and so we will cease to distinguish between them. This action is the data of an oplax monoidal functor
풬 → Fun(sect(ΨBor), sect(ΨBor)).
One can describe this action more explicitly as follows. Let 퐴 ∈ sect(ΨBor) and 퐺 ∈ 풬. By the formula of Observation 3.3.5, we see that the action of 퐺 on 퐴 yields a new section 퐴퐺 whose value on a groupoid 퐵퐻 is given by
퐺 퐺 퐴 (퐵퐻) = 훽퐵퐻 Φ 푗퐵퐻×퐵퐺퐴(퐵퐻 × 퐵퐺).
Glo If 퐴 is in the full subcategory CAlgBor ⊂ sect(ΨBor) of Definition 3.4.5, then this is equivalent to 퐴휏퐺 with the trivial 퐻-action for all 퐻; thus, if 퐴 is a Borel global 퐺 algebra, then 퐴 is as well. It follows that the oplax action on sect(ΨBor) restricts Glo to an oplax action on CAlgBor. Unwinding the definitions, we see that the maps in 풬 act as described in the statement of Theorem 3.2.7.
Glo Theorem 3.4.8. The restriction functor CAlgBor → CAlg induces an equivalence of ∞-categories.
Proof. The relevant restriction functor is implemented by restricting a section of ΨBor along the inclusion 푖 : Span(Fin) → Glo (cf. Remark 3.3.4). We show this induces an equivalence by proving the existence of and computing the ΨBor-right Kan extension along this inclusion.
46 Let 퐴 ∈ CAlg be an E∞-ring, and let us first compute the pointwise ΨBor-right Kan extension in the diagram
퐴 Span(Fin) GloSpBor
ΨBor Glo Glo at 푋 ∈ Glo; this is given by a certain ΨBor-limit indexed by the 2-category
Span(Fin) ×Glo Glo푋/ .
We first examine the indexing category. An object of Span(Fin) ×Glo Glo푋/ is a span of groupoids (푋 ← 푌 → 푍) such that 푍 is a finite set and the map 푌 → 푍 has discrete fibers. It follows that 푌 must also be a finite set. We shall use the shorthand Fin/푋 for the category Fin ×Gpd Gpd/푋 whose objects are finite sets 푇 equipped with a map of groupoids 푝 : 푇 → 푋. We have the following lemma:
op Lemma 3.4.9. The functor 휃 : (Fin/푋 ) → Span(Fin) ×Glo Glo푋/ defined by
푝 (푝 : 푇 → 푋) ↦→ (푋 ←− 푇 −→= 푇 ) admits a right adjoint.
푞 Proof. The adjoint is given by sending (푋 ←− 푇 → 푈) to the map 푞 : 푇 → 푋. The groupoid 푇 is a finite set by the remarks above and it is immediate that thisisa right adjoint.
It follows that 휃 is coinitial and we have reduced to computing the relative op limit over (Fin/푋 ) . Unwinding the definitions, a limit of this diagram relative to ˜ the fibration ΨBor is the data of 퐴 ∈ Fun(푋, Sp) equipped with compatible maps ˜ op 휈푝 : 퐴(푇 ) → 휃(푝)*퐴 for each 푝 : 푇 → 푋 ∈ Fin/푋 such that for any 퐸 ∈ Fun(푋, Sp), the natural map
Hom (퐴,˜ 퐸) −→ lim Hom (휃(푝) 퐴,˜ 휃(푝) 퐸) Fun(푋,Sp) op Fun(푇,Sp) * * 푝:푇 →푋∈Fin/푋 −→ lim Hom (퐴(푇 ), 휃(푝) 퐸) op Fun(푇,Sp) * 푝:푇 →푋∈Fin/푋 is an equivalence.
47 ˜ = 푋 We claim that 퐴 := (* ← 푋 −→ 푋)*퐴(*) = triv 퐴(*) (together with the obvious choice of 휈푝) has the desired universal property. Because everything in sight sends disjoint unions in 푋 to products, it suffices to consider the case when 푋 is connected. Without loss of generality, let 푋 = 퐵퐺. Then we have Fin/푋 ≃ free Fin퐺 , the category of finite free 퐺-sets, and we wish to show that for any 퐸 ∈ Fun(퐵퐺, Sp), the natural map
퐺 HomFun(퐵퐺,Sp)(triv (퐴(*)),퐸)
→ lim HomFun(푈/퐺,Sp)(퐴(푈/퐺), (퐵퐺 ← 푈/퐺 → 푈/퐺)*퐸) free op 푈∈(Fin퐺 ) is an equivalence. Since 퐴 was assumed to be an E∞-algebra, the functor of 푈 on free the right-hand side is product preserving. Since Fin퐺 is generated freely under coproducts by the full subcategory on the transitive free 퐺-set, it follows that the limit diagram is right Kan extended from that subcategory, and so we have
lim HomFun(푈/퐺,Sp)(퐴(푈/퐺), (퐵퐺 ← 푈/퐺 → 푈/퐺)*퐸) free op 푈∈(Fin퐺 )
≃ lim HomSp(퐴(*), (퐵퐺 ← * → *)*퐸) 퐵퐺 푒 ℎ퐺 ≃ HomSp(퐴(*), res퐺 퐸) 퐺 ≃ HomFun(퐵퐺,Sp)(triv (퐴(*)), 퐸), as desired.
We have shown that the ΨBor-right Kan extension exists at every point, and so by [28, Lemma 4.3.2.13], the ΨBor-right Kan extension exists. Moreover, our calculation shows that this Kan extension takes an E∞-algebra to a Borel global algebra with the same underlying E∞-ring. It follows that both the unit and the counit of the resulting adjunction are equivalences. Thus, the left adjoint induces Glo an equivalence of ∞-categories CAlgBor → CAlg as desired.
This concludes the proof of Theorem 3.2.7. We make two remarks about the proof:
Remark 3.4.10. In fact, the proof of Proposition 3.3.7 shows that the monoidal functor from 풬 arises from one defined on the larger 2-category 풬̃︀. We do not know if this additional generality has interesting consequences and will not use it in this paper.
48 Remark 3.4.11. The action of Theorem 3.2.7 could not have been produced di- rectly using Proposition A.0.10 because the coCartesian fibration ΨBor does not arise as the restriction of a fibration over all of Glo+. This results in an oplax, rather than strict, action. In terms of equivariant homotopy theory, this cor- responds to the fact that Borel equivariant homotopy theory does not admit a monoidal fixed point functor analogous to geometric fixed points in genuine equiv- ariant homotopy theory.
49 50 Chapter 4
Partial Algebraic 퐾-theory
Let 풞 be an exact category in the sense of Quillen [38]. Then, the zeroth 퐾-theory of 풞, denoted 퐾0(풞), is the free abelian group on the objects of 풞 subject to the relation [퐴] + [퐶] = [퐵] for every short exact sequence 0 → 퐴 → 퐵 → 퐶 → 0 in 풞. Quillen categorified this construction, defining the higher algebraic 퐾-theory space 퐾(풞) by means of a certain category 푄(풞). Waldhausen [45] generalized the construction of algebraic 퐾-theory to what are now known as “Waldhausen categories” by means of his 푆∙-construction and proved that the definition coincides with Quillen’s in the special case of an exact category. These constructions were generalized to the higher categorical setting by Barwick [5, 7, 8]. The goal of this chapter is to give analogs of these constructions in a non-group- complete setting. In §4.1, we introduce a construction called partial 퐾-theory, which is a non-group-complete analog of algebraic 퐾-theory. It associates to a part Waldhausen ∞-category 풞 a (not necessarily grouplike) E∞-space 퐾 (풞) with the following two properties:
part gp 1. There is a canonical equivalence of E∞-spaces 퐾 (풞) ≃ 퐾(풞) (Corollary 4.1.9.2).
part 2. The monoid 휋0(퐾 (풞)) is the free (discrete) monoid on 풞 subject to the relation [퐴]+[퐶] = [퐵] for every short exact sequence 0 → 퐴 → 퐵 → 퐶 → 0 (Proposition 4.1.10).
Then, in §4.2, we give an alternate construction of partial 퐾-theory for exact ∞- categories via Quillen’s 푄-construction and show that it coincides with the previous definition (Theorem 4.2.3).
51 4.1 Partial 퐾-theory via the 푆∙-construction
Let 풞 be a Waldhausen ∞-category in the sense of [8]. One can extract from 풞 a simplicial ∞-category 푆∙(풞) such that 푆푛(풞) is equivalent to the ∞-category of sequences of cofibrations
* ˓→ 푋1 ˓→ 푋2 ˓→ · · · ˓→ 푋푛 between objects 푋푖 ∈ 풞 ([45, 8]).
Definition 4.1.1 ([45, 8]). The algebraic 퐾-theory of 풞 is the E1-space
≃ 퐾(풞) := Ω|푆∙(풞) |,
≃ where 푆∙(풞) denotes the simplicial space obtained by taking the maximal sub- groupoid of 푆∙(풞) level-wise.
The algebraic 퐾-theory of 풞 can be thought of as the universal way to make 푆∙(풞) into a grouplike E1-monoid in spaces. As explained in the introduction, we need a variant of this construction which can produce non-group-complete monoids.
Definition 4.1.2. A Segal space is a functor 푋(−) : ∆op → 풮 such that for each 푛 ≥ 1, the collection of maps 휌푖 : [1] → [푛] in ∆ defined by 휌푖(0) = 푖, 휌푖(1) = 푖 + 1 for 0 ≤ 푖 ≤ 푛 − 1 induces an equivalence
푛−1 ∏︁ 푛 푋(휌푖): 푋([푛]) ≃ 푋([1]) . 푖=0
We will denote the ∞-category of Segal spaces by Seg(풮).
Definition/Proposition 4.1.3 ([28], Proposition 4.1.2.10). Let Mon(풮) denote the ∞-category of E1-monoids in spaces. Then there is a fully faithful functor
op B : Mon(풮) → Fun(∆ , 풮) which sends a monoid 푀 to its bar construction
(︀ )︀ B푀 = * 푀 푀 × 푀 ··· .
52 The essential image of B is the full subcategory of Segal spaces 푋 with the ad- ditional property that 푋([0]) ≃ *. We will sometimes implicitly identify Mon(풮) with this subcategory of simplicial spaces.
Definition 4.1.4. Since the full subcategory Mon(풮) ⊂ Fun(∆op, 풮) is closed under limits and filtered colimits, the functor B admits a left adjoint [29, Corollary 5.5.2.9], which we denote by
op L : Fun(∆ , 풮) → Mon(풮).
The main object of study in this chapter is:
Definition 4.1.5. Let 풞 be a Waldhausen ∞-category. Then the partial algebraic K-theory of 풞 is the E1-monoidal space
part ≃ 퐾 (풞) := L(푆∙(풞) ).
It can be helpful to rephrase this definition in terms of the notion of complete Segal spaces, which we now very briefly recall.
Recollection 4.1.6. The ∞-category of small ∞-categories Cat∞ can be identified with a full subcategory CplSeg(풮) ⊂ Seg(풮) of Segal spaces known as complete Segal spaces [26, Corollary 4.3.16] and due op to [39]. Via this identification, the inclusion Cat∞ ⊂ Fun(∆ , 풮) admits a left adjoint, which we denote by
op CSS : Fun(∆ , 풮) → Cat∞ .
The functor CSS can be described as the unique colimit preserving functor sending the standard 푛-simplex as a simplicial space to the standard 푛-simplex as an ∞- category. Just as the subcategory CplSeg(풮) ⊂ Fun(∆op, 풮) of complete Segal spaces can op be identified with Cat∞, the ∞-category Seg(풮) ⊂ Fun(∆ , 풮) can be identified fl with the ∞-category Cat∞ of flagged ∞-categories [3, Theorem 0.26]. A flagged ∞-category is a triple (풞, 푋, 푓) consisting of an ∞-category 풞, a space 푋, and an essentially surjective functor 푓 : 푋 → 풞. A Segal space 푌∙ determines a flagged
53 ∞-category via the canonical functor 푌0 → CSS(푌∙). Under this identification, the full subcategory CplSeg(풮) ⊂ Seg(풮) corresponds to the full subcategory of flagged ∞-categories (풞, 푋, 푓) with the property that 푓 induces an equivalence of spaces 푋 ≃ 풞≃. The left adjoint to this inclusion corresponds to the forgetful fl functor Cat∞ → Cat∞ given by (풞, 푋, 푓) ↦→ 풞.
Remark 4.1.7. Combining Definition/Proposition 4.1.3 with Recollection 4.1.6, we obtain an equivalence of ∞-categories between Mon(풮) and the ∞-category (Cat∞)0 of ∞-categories equipped with an essential surjection from a point. This equivalence can be thought of as sending an E1-monoid 푀 to the ∞-category 퐵푀 with one object whose space of endomorphisms is 푀. Since the above constructions are all compatible with finite products, this equivalence also lifts to an equivalence, for any 푛 ≥ 1, between E푛-monoids in spaces and E푛−1-monoidal ∞-categories with the property that the inclusion of the unit is an essential surjection.
Using this language, Definition 4.1.5 can be rephrased as follows:
Proposition 4.1.8. Let 풞 be a Waldhausen ∞-category. Then there is an equiv- alence of ∞-categories ≃ part CSS(푆∙(풞) ) ≃ 퐵퐾 (풞).
≃ In other words, just as 퐵퐾(풞) is the underlying space of 푆∙(풞) , the ∞- part ≃ category 퐵퐾 (풞) is the “underlying ∞-category” of 푆∙(풞) . We prove Proposi- tion 4.1.8 at the end of the section. One consequence of Proposition 4.1.8 is that although partial 퐾-theory has a universal property as an E1-space, it naturally admits the structure of an E∞-space.
op Lemma 4.1.9. The functor CSS : Fun(∆ , 풮) → Cat∞ commutes with finite products.
Proof. We would like to show that for simplicial spaces 푋 and 푌 , the natural map
CSS(푋 × 푌 ) → CSS(푋) × CSS(푌 ) is an equivalence. We observe that products preserve colimits separately in each op variable in both Fun(∆ , 풮) and Cat∞, and the functor CSS preserves colim- its. Consequently, it suffices to check this statement on representable objects in
54 Fun(∆op, 풮). But each representable simplicial space ∆푛 is in the image of the op fully faithful right adjoint Cat∞ ⊂ Fun(∆ , 풮), which clearly preserves products, so the conclusion follows.
≃ Since the coproduct on 풞 endows 푆∙(풞) with the structure of an E∞-monoid ≃ in simplicial spaces, we may use Lemma 4.1.9 to equip CSS(푆∙(풞) ) with the structure of a symmetric monoidal ∞-category. We then have the following two corollaries of Proposition 4.1.8.
Corollary 4.1.9.1. Let 풞 be a Waldhausen ∞-category. Then the coproduct on part 풞 endows 퐾 (풞) with the structure of an E∞-monoidal space.
part gp Corollary 4.1.9.2. There is a natural equivalence 퐾 (풞) ≃ 퐾(풞) of E∞- spaces.
Proof. Note that the functor 풮 → Fun(∆op, 풮) sending a space to the constant simplicial space factors as a composite
풮 → CplSeg(풮) ⊂ Fun(∆op, 풮) through complete Segal spaces. Taking left adjoints and looping, we obtain an equivalence of E1-spaces
part gp ≃ ≃ ≃ (퐾 (풞)) ≃ Hom|CSS(푆∙(풞) )|(*, *) ≃ Ω|CSS(푆∙(풞) )| ≃ Ω|푆∙(풞) | ≃ 퐾(풞).
We have seen that the relevant left adjoints commute with finite products, so this is in fact an equivalence of E∞-spaces.
part As in the case of ordinary 퐾-theory, one can explicitly describe 퐾0 (풞):
Proposition 4.1.10. Let 풞 be a Waldhausen ∞-category. Then the monoid
part part 퐾0 (풞) := 휋0퐾 (풞) is freely generated by the objects of 풞 modulo the relation [퐴]+[퐶] = [퐵] for every short exact sequence 0 → 퐴 → 퐵 → 퐶 → 0.
Proof. Let Mon(Set) denote the category of monoids in sets and let
op B0 : Mon(Set) → Fun(∆ , Set)
55 denote the functor which sends a monoid to its bar construction. Then, B0 factors through the functor
op op 푖* : Fun(∆≤2, Set) → Fun(∆ , Set)
op op given by right Kan extension along the inclusion 푖 : ∆≤2 → ∆ of the full subcat- egory spanned by the objects [0], [1], and [2]. We obtain a commutative diagram of right adjoints Mon(풮) B Fun(∆op, 풮)
Mon(Set) B0 Fun(∆op, Set)
푖* B0 op Fun(∆≤2, Set).
All of the functors in the diagram have left adjoints; the left adjoint of the * upper vertical arrows are by taking 휋0, the left adjoint of 푖* is the restriction 푖 , and we let L0 denote the left adjoint to B0. Using the commutative diagram of left adjoints, we deduce that for a simplicial space 푋, there is an isomorphism
∼ * 휋0L푋 = L0푖 휋0푋.
≃ We would like to compute this in the case that 푋 = 푆∙(풞) and show that it has the proposed description. Let 푀0 be a monoid in sets. Unwinding the definitions, the data of a map * ≃ 푖 휋0(푆∙(풞) ) → B0(푀0) op in Fun(∆≤2, Set) is exactly the data of an object in 푀0 for each equivalence class of object in 풞 satisfying the usual additivity condition in exact sequences. It follows * ≃ that L0푖 휋0(푆∙(풞) ) satisfies the required universal property.
Remark 4.1.11. In a stable setting, partial 퐾-theory does not produce anything new. For instance, if 풞 is a stable ∞-category, one has for every 푋 ∈ 풞 a cofiber part sequence 푋 → 0 → Σ푋. It follows from Proposition 4.1.10 that in 퐾0 (풞), [푋] part has an inverse given by [Σ푋]. Consequently, 퐾0 (풞) is group complete and the natural map 퐾part(풞) → 퐾(풞) is an equivalence by Corollary 4.1.9.2.
We now return to the proof of Proposition 4.1.8.
56 Proof of Proposition 4.1.8.
Notation. In the course of this proof, if 푓 is a fully faithful functor, we will denote by 푓 퐿 (resp. 푓 푅) its left (resp. right) adjoint, provided it exists. op op Let Fun*(∆ , 풮) ⊂ Fun(∆ , 풮) be the full subcategory of simplicial spaces 푋∙ op such that 푋0 ≃ *. Then, the inclusion Mon(풮) ⊂ Fun(∆ , 풮) factors through an op ≃ inclusion 푘 : Mon(풮) ˓→ Fun*(∆ , 풮). Because 푆∙(풞) is in the full subcategory op Fun*(∆ , 풮), there is an equivalence
≃ 퐿 ≃ L푆∙(풞) ≃ 푘 푆∙(풞) . (4.1)
Since * ∈ Fun(∆op, 풮) is left Kan extended from its value at [0], it is an initial op object in Fun*(∆ , 풮). It follows that there is a fully faithful functor embedding
op op 푖0 : Fun*(∆ , 풮) → Fun(∆ , 풮)*/.
This extends to a commutative diagram of fully faithful functors
푗1 푗0 op CplSeg(풮)*/ Seg(풮)*/ Fun(∆ , 풮)*/
푖1 푖0
푘 op Mon(풮) Fun*(∆ , 풮).
푅 We note that 푖0 admits a right adjoint 푖0 which extracts the simplices which only involve the given zero simplex. It is given on a simplicial space 푇∙ by the formula 푅 ×푛+1 (푖0 푇∙)푛 ≃ 푇푛 ×(푇0) *,
×푛+1 where 푇푛 → 푇0 are the vertex maps. It is immediate from this formula that 푅 op 푖0 takes the full subcategory Seg(풮)*/ ⊂ Fun(∆ , 풮)*/ to the full subcategory op 푅 푅 푅 Mon(풮) ⊂ Fun*(∆ , 풮), so 푖1 admits a right adjoint 푖1 such that 푖0 푗0 ≃ 푘푖1 . We now diagram chase:
퐿 ≃ 푅 퐿 ≃ 푅 퐿 ≃ 푅 퐿 퐿 ≃ 푘 (푆∙(풞) ) ≃ 푖1 푖1푘 (푆∙(풞) ) ≃ 푖1 푗0 푖0(푆∙(풞) ) ≃ 푖1 푗1푗1 푗0 푖0(푆∙(풞) ). (4.2)
퐿 퐿 ≃ 퐿 ≃ ≃ Finally, we note that 푗1 푗0 푖0(푆∙(풞) ) = (푗0푗1) 푖0(푆∙(풞) ) is exactly CSS(푆∙(풞) ) 푅 with the canonical basepoint, and the functor 푖1 푗1 takes a pointed ∞-category to the endomorphisms of the basepoint. Combining this with the equivalences (4.1)
57 and (4.2) yields the result.
4.2 Partial K-theory via the 푄-construction
Throughout this section, let 풞 be an exact ∞-category in the sense of [5, Definition 1.3]. Then, one can form an ∞-category 푄 풞 known as the 푄-construction on 풞 [5, Definition 3.8].
Example 4.2.1. When 풞 = VectF푝 is the category of finite dimensional F푝-vector spaces, 푄 풞 is the ordinary category whose objects are finite dimensional F푝-vector 푉 Hom (푈, 푉 ) spaces and where 푄VectF푝 is the set of isomorphism classes of spans 푈 푊 ˓→ 푉 where the backward arrow is surjective and the forward arrow is injective. Composition is given by the usual composition of spans.
Quillen [38] defined the 퐾-theory space of 풞 as Ω|푄 풞 |. On the other hand, 풞 can be regarded as a Waldhausen ∞-category and one can consider its 퐾- theory in the sense of Definition 4.1.1. The following theorem asserts that these constructions agree:
Theorem 4.2.2 ([45] §1.9, [5] Proposition 3.7). There is an equivalence of spaces
≃ |푄 풞 | ≃ |푆∙(풞) |.
We now describe how to make a pre-group-completed variant of this construc- tion. Thinking of the space |푄 풞 | as an ∞-category, the natural map 푄 풞 ↦→ |푄 풞 | can be thought of as formally inverting all the morphisms. On the other hand, the 퐾-theory of 풞 arises as the endomorphisms of the unit object in |푄 풞 |. Accordingly, to create a version of 퐾-theory which is not group complete, one could instead contemplate inverting only some of the morphisms of 푄 풞. Let ℒ ⊂ Mor(푄 풞) denote the backward (left-pointing) arrows, i.e., those of the form 푋 푌 → 푌 . The localization 푄 풞[ℒ−1] is a symmetric monoidal ∞-category which comes with a canonical point represented by 0 ∈ 풞. The endomorphisms of 0 in 푄 풞[ℒ−1] is then a monoid in spaces which group completes to 퐾(풞). The main theorem of this section is that this monoid coincides with the partial 퐾-theory of 풞:
58 Theorem 4.2.3. Let 풞 be an exact ∞-category. Then, regarding 풞 as a Wald- hausen ∞-category as in [6, Corollary 4.8.1], there is an equivalence of E1-spaces
part 퐾 (풞) ≃ Hom푄 풞[ℒ−1](0, 0).
This is a categorified analog of the theorems of Waldhausen and Barwick re- lating 퐾-theory via the 푆∙-construction to 퐾-theory via the 푄-construction. We will prove Theorem 4.2.3 in a more precise form as Corollary 4.2.12.1. We begin by reviewing the notion of edgewise subdivision.
Definition 4.2.4. Let 휖 : ∆ → ∆ be the functor which takes a linearly ordered set 퐼 to the join 퐼op ⋆ 퐼. Given an ∞-category 풟 and a simplicial object 푇 ∈ Fun(∆op, 풟), one can form a new simplicial object 휖*푇 by precomposition with 휖, which we will refer to as the edgewise subdivision of 푇 . This construction comes equipped with two natural maps induced by the inclusions 퐼 ⊂ 퐼op ⋆ 퐼 and 표푝 op * op * op 퐼 ⊂ 퐼 ⋆ 퐼, which we denote by 휂푇 : 휖 푇 → 푇 and 휂푇 : 휖 푇 → 푇 , respectively.
Example 4.2.5. When 풟 = Set and 푇 is a quasicategory, 휖*푇 is also a quasi- category and presents the twisted arrow category of 푇 ([28, Proposition 5.2.1.3], beware the opposite convention for morphism direction).
We will be particularly interested in the following example, which says roughly that Quillen’s 푄-construction arises from Waldhausen’s 푆∙-construction by edge- wise subdivision:
≃ Example 4.2.6. When 풟 = 풮 and 푇 = 푆∙(풞) for an exact ∞-category 풞, there is an equivalence of ∞-categories
* ≃ 푄 풞 ≃ CSS(휖 푆∙(풞) ).
This follows from combining [5, Proposition 3.4] and [5, Proposition 3.7].
Recall that 퐾part(풞) is defined as the endomorphisms of the unit object in ≃ CSS(푆∙(풞) ); accordingly, Theorem 4.2.3 asserts a relationship between the sim- ≃ * ≃ plicial spaces 푆∙(풞) and 휖 푆∙(풞) . In the setting of ordinary 퐾-theory (Theorem 4.2.2), one needs to compare these simplicial spaces at the level of geometric re- alization; this boils down to the classical fact that for any simplicial space 푇 , the map 휂푇 becomes an equivalence after passing to underlying spaces [41, A.1]. The
59 proof of Theorem 4.2.3 refines this to a statement about underlying ∞-categories. Namely, instead of passing all the way to underlying spaces, one can study the functor 휂푇 after applying CSS. The main technical result of this section is that * while the resulting functor CSS(휂푇 ) : CSS(휖 푇 ) → CSS(푇 ) is not generally an equivalence, it can be described as a localization at a particular collection of mor- phisms.
Remark 4.2.7. Let 퐾 be a simplicial set which is a quasicategory (i.e., fibrant op in the Joyal model structure), and let 퐾풮 ∈ Fun(∆ , 풮) denote 퐾 regarded as a discrete simplicial space. It follows from [21, Theorem 4.11] that there is a natural equivalence of ∞-categories 퐾 ≃ CSS(퐾풮 ).
We first study the case of when 푇 is a standard simplex.
* 푛 Example 4.2.8. By Example 4.2.5 and Remark 4.2.7, the ∞-category CSS(휖 ∆풮 ) is the twisted arrow category of ∆푛: 푛푛
. . .. .
22 ··· 2푛 (4.3)
11 12 ··· 1푛
00 01 02 ··· 0푛.
With reference to diagram (4.3), the functor
* 푛 푛 푛 CSS(휂 푛 ) : CSS(휖 ∆ ) → CSS(∆ ) ≃ ∆ Δ풮 풮 풮 projects down to the horizontal axis.
푛 * 푛 Definition 4.2.9. Let ℒ(∆풮 ) denote the subset of the morphisms of CSS(휖 ∆풮 ) CSS(휂 푛 ) whose images under Δ풮 are homotopic to identity morphisms. These cor- respond to vertical maps in diagram (4.3).
Lemma 4.2.10. The functor
* 푛 푛 푛 CSS(휂 푛 ) : CSS(휖 ∆ ) → CSS(∆ ) ≃ ∆ Δ풮 풮 풮
60 푛 * 푛 푛 exhibits ∆ as the localization of CSS(휖 ∆풮 ) at the collection of morphisms ℒ(∆풮 ).
Proof. We will refer to diagram (4.3) of Example 4.2.8. By Remark 4.2.7, the functor of the lemma is identified with the natural functor of ∞-categories 휂Δ푛 : 휖*∆푛 → ∆푛. Regarding 휖*∆푛 as a marked simplicial set by marking the morphisms 푛 * 푛 푛 ♭ in ℒ(∆풮 ), it suffices to show that 휂Δ푛 : 휖 ∆ → (∆ ) is a weak equivalence in the marked model structure, where (∆푛)♭ denotes the simplicial set ∆푛 with only the degenerate edges marked (cf. [29, Proposition 3.1.3.7], [19, 1.1.3]). Consider the sequence of maps 휂 푛 (∆푛)♭ −→푖 휖*∆푛 −−→Δ (∆푛)♭ where 푖 includes the bottom edge 00 → 01 → 02 → · · · → 0푛 of (4.3). The composite is the identity, so the lemma follows by observing that the inclusion 푖 is a marked anodyne extension ([29, Remark 3.1.3.4]).
Definition 4.2.11. Let 푇 be a simplicial space. With reference to Definition 4.2.9, define the subset ℒ(푇 ) ⊂ Mor(CSS(휖*푇 )) by
* ′ 푛 ′ 푛 ℒ(푇 ) := {훾 | 훾 = CSS(휖 푓)훾 for some morphism 푓 : ∆풮 → 푇 and edge 훾 ∈ ℒ(∆풮 )}.
We can now state and prove the main technical result.
Proposition 4.2.12. Let 푇 be a simplicial space. Then the functor
* * CSS(휖 휂푇 ) : CSS(휖 푇 ) → CSS(푇 ) exhibits CSS(푇 ) as the localization of CSS(휖*푇 ) at the morphisms in ℒ(푇 ).
Proof. We would like to show that for any ∞-category 풟, the induced map
Fun(CSS(푇 ), 풟) → Fun(CSS(휖*푇 ), 풟) is the inclusion of the full subcategory of functors which send the morphisms in ℒ(푇 ) to equivalences in 풟. Write the simplicial space 푇 as a colimit of representables:
푇 = colim ∆푛 . 푛 풮 Δ풮 →푇
61 Then, since 휖* and CSS preserve colimits, we have the commutative square
Fun(CSS(푇 ), 풟) Fun(CSS(휖*푇 ), 풟) ∼ ∼ lim Fun(CSS(∆푛 ), 풟) lim Fun(CSS(휖*∆푛 ), 풟). 푛 풮 푛 풮 Δ풮 →푇 Δ풮 →푇
By Lemma 4.2.10, the bottom horizontal arrow is the inclusion of precisely the 푛 푛 subcategory of functors which invert ℒ(∆풮 ) for each ∆풮 → 푇 , and the proposition follows.
We apply Proposition 4.2.12 in the situation of Example 4.2.6. Under the equivalence * ≃ 푄 풞 ≃ CSS(휖 푆∙(풞) ),
≃ the subset ℒ(푆∙(풞) ) of morphisms corresponds to the collection ℒ of backward arrows of 푄 풞 (i.e., those of the form 푋 푌 → 푌 ). We therefore have the following corollary of Proposition 4.2.12, which completes the proof of Theorem 4.2.3:
≃ Corollary 4.2.12.1. There is a functor 푄 풞 → CSS(푆∙(풞) ) which extends to an equivalence of ∞-categories
−1 ≃ 푄 풞[ℒ ] ≃ CSS(푆∙(풞) ), where ℒ denotes the collection of backward morphisms. Taking endomorphisms of the zero object on both sides and applying Proposition 4.1.8, we obtain an equivalence of E1-spaces
part 퐾 (풞) ≃ Hom푄 풞[ℒ−1](0, 0).
62 Chapter 5
The Partial 퐾-theory of F푝
part We will see in §6 that the monoidal ∞-category 퐵퐾 (F푝) acts on the ∞-category 퐹 CAlg푝 of 퐹푝-stable E∞-rings. Our goal in this chapter is to give a computation of part 퐾 (F푝) up to 푝-completion. To motivate the result, recall the following theorem of Quillen:
Theorem 5.0.1 (Quillen [37]). The natural map
∼ 퐾(F푝) → 휋0퐾(F푝) = Z induces an isomorphism in F푝-homology.
In particular, the F푝-homology of 퐾(F푝) is trivial in positive degrees. The main result of this chapter is the analog of Quillen’s theorem for partial 퐾-theory:
Theorem 5.0.2. The natural map
part part ∼ 퐾 (F푝) → 휋0퐾 (F푝) = Z≥0 induces an isomorphism in F푝-homology.
In §5.1, we provide a formula for partial 퐾-theory in terms of a colimit of part spaces in the 푆∙-construction. Then in §5.2, we specialize to the case of 퐾 (F푝) and evaluate this formula up to F푝-homology equivalence.
63 5.1 Computing partial 퐾-theory
In this section, we give a formula for the functor L of Definition 4.1.4 in terms of a certain colimit. To do so, we show that E1-spaces can be presented via an ∞- categorical Lawvere theory (Proposition 5.1.4). We then reinterpret B as a certain restriction map and L as the corresponding left Kan extension.
Definition 5.1.1. Let 풯퐴 denote the opposite category of the full subcategory of (discrete, associative) monoids spanned by those which are free and finitely generated. For a finite set 푆, we will denote the free monoid on 푆 by Free(푆). We will refer to 풯퐴 as the theory of associative monoids. Notation 5.1.2. For a linearly ordered set 푆, let 푆± denote the linearly ordered set {−∞} ∪ 푆 ∪ {∞}. It will be convenient to think of ∆op as the category of (possibly empty) linearly ordered sets 푆, where morphisms from 푆 to 푇 are order preserving maps 푆± → 푇 ± preserving ±∞. This description arises by associating to a finite linearly ordered set in ∆ its corresponding linearly ordered set of “gaps,” or pairs of adjacent elements. We will denote the linearly ordered set of gaps in [푛] ∈ ∆ by (푛) ∈ ∆op, so that (푛) has 푛 elements, which we call 1, 2, ··· , 푛, and (푛) is always implicitly regarded as an object in ∆op. Definition 5.1.3. Define a functor
op 퐹 : ∆ → 풯퐴 by (푛) ↦→ Free((푛)) on objects, and by sending the morphism 푓 :(푛) → (푚) in ∆op to the morphism 퐹 (푓) : Free((푚)) → Free((푛)) which sends the generator corresponding to 푖 ∈ (푚) to the product of the genera- tors corresponding to 푓 −1(푖), using the order from (푛). We now show the Lawvere theory presentation of monoids suggested by Defini- tion 5.1.1 coincides with the notion of monoid from Definition/Proposition 4.1.3. * op Proposition 5.1.4. The restriction functor 퐹 : Fun(풯퐴, 풮) → Fun(∆ , 풮) sends × the full subcategory Fun (풯퐴, 풮) ⊂ Fun(풯퐴, 풮) of product preserving functors to the full subcategory Mon(풮) ⊂ Fun(∆op, 풮) of monoids in spaces. Moreover, the restricted functor * × 퐹 : Fun (풯퐴, 풮) → Mon(풮)
64 is an equivalence of ∞-categories.
Proof. The first statement is clear so we focus on the second. Let 푀 : ∆op → 풮 be a monoid. We shall compute the right Kan extension along the functor 퐹 and see that it determines an inverse equivalence. op The right Kan extension at Free((푛)) ∈ 풯퐴 is indexed by the category ∆ ×풯퐴 op (풯퐴)Free((푛))/ whose objects are pairs ((푚), 푓) where (푚) ∈ ∆ and 푑 : Free((푚)) → Free((푛)) is a map of monoids. We will show that the value of the right Kan extension at Free((푛)) is 푀 푛. We first describe in detail the special case 푛 = 1. In this case, the map 푑 : Free((푚)) → Free((1)) ≃ Z≥0 assigns a nonnegative integer, which we can think of as a “degree,” to each of the generators of Free((푚)). The morphisms in op ∆ ×풯퐴 (풯퐴)Free((푛))/ from ((푚1), 푑1) to ((푚2), 푑2) are maps 푓 :(푚1) → (푚2) in ∆op such that 퐹 (푓) preserves the degree.
op Observation. Consider the full subcategory 풥1 ⊂ ∆ ×풯퐴 (풯퐴)Free((1))/ spanned by those ((푚), 푑) with the property that 푑 sends every generator of Free((푚)) to 1. Then the inclusion of 풥1 admits a right adjoint. Explicitly, let 푥1, 푥2, ··· , 푥푚 ∈ Free((푚)) denote the generators and 푑 : Free((푚)) → Z≥0 be any map of monoids; then the right adjoint sends ((푚), 푑) to the unique object in 풥1 whose underlying monoid is Free((푑(푥1) + 푑(푥2) + ··· + 푑(푥푚))).
It follows that the subcategory 풥1 is coinitial. One can (and we will) view 풥1 op in an alternate way, as the wide subcategory of ∆ whose morphisms from (푚1) ± ± to (푚2) are those determined by maps of linearly ordered sets (푚1) → (푚2) ± which are isomorphisms when restricted to the preimage of (푚2) ⊂ (푚2) . With this notation, the value of the desired right Kan extension is computed as
lim 푀 푚. (푚)∈풥1
This diagram of spaces is right Kan extended from the full subcategory of 풥1 spanned by the object (1) ∈ 풥1, and so the limit is given simply by 푀, as desired. For a general 푛, one can think of the function 푑 : Free((푚)) → Free((푛)) as assigning a “generalized degree” which takes values in Free((푛)) instead of just Z≥0 = Free((1)). One then extracts the coinitial subcategory 풥푚 determined by those 푑 which assign to each generator an element of “generalized degree one” in the sense that they send generators of Free((푚)) to generators of Free((푛)). The
65 resulting diagram is right Kan extended from the full subcategory of 풥푚 spanned by objects whose underlying monoid is free on one generator; there are 푛 of these (one for each generator of Free((푛))) and no maps between them, so the value of the limit is 푀 푛, as desired. It follows from the above calculation that the resulting functor is product preserving and that the unit and counit maps induce equivalences.
Under the identification of Proposition 5.1.4, the composite
× op Mon(푆) ≃ Fun (풯퐴, 풮) → Fun(∆ , 풮) is exactly the functor B of Definition/Proposition 4.1.3. But the target of B also admits a description as an ∞-category of product preserving functors.
Notation 5.1.5. Let (∆op)× denote the free product completion of ∆op. It has the universal property that for any ∞-category 풞 which has finite products, re- striction along the inclusion ∆op → (∆op)× induces an equivalence of ∞-categories Fun×((∆op)×, 풞) ≃ Fun(∆op, 풞). Explicitly, the objects of (∆op)× are finite sets 푆 together with a collection {(푛푠)}푠∈푆 of finite linearly ordered sets indexed by 푆. A morphism from {(푛푠)}푠∈푆 to {(푚푡)}푡∈푇 is the data of a function 휑 : 푇 → 푆 together with a morphism op (푛휑(푡)) → (푚푡) in ∆ for each 푡 ∈ 푇 .
op Since the category 풯퐴 admits finite products, the functor 퐹 : ∆ → 풯퐴 ex- × op × tends uniquely to a product preserving functor 퐹 : (∆ ) → 풯퐴. Under this identification, B is given by restriction along 퐹 ×:
× * × op × × B = (퐹 ) : Fun ((∆ ) , 풮) → Fun (풯퐴, 풮).
Its left adjoint L will be given by left Kan extension, and will be computed by a colimit over a certain indexing category that we now discuss:
Definition 5.1.6. We define a category ℐ as follows. The objects of ℐ are tuples (퐼, 퐽, 푓) where 퐼 and 퐽 are (possibly empty) finite linearly ordered sets and 푓 : 퐼 퐽 is an order preserving surjection. We think of 푓 as partitioning 퐼 into convex sub- sets indexed by 퐽. As such, if we denote the elements of 퐽 by 푗1 < 푗2 < ··· < 푗푚, −1 −1 −1 then we will refer to a typical element (퐼, 퐽, 푓) ∈ ℐ by (푓 푗1)(푓 푗2) ··· (푓 푗푚), −1 −1 −1 or more informally by (|푓 (푗1)|)(|푓 (푗2)|) ··· (|푓 (푗푚)|).
66 The morphisms in ℐ from (퐼, 퐽, 푓) to (퐼′, 퐽 ′, 푓 ′) are given by commuting squares of order preserving maps: 퐼 퐼′
푓 푓 ′ 퐽 퐽 ′, which we think of as a map of linearly ordered sets 퐼 → 퐼′ that refines the parti- tions. As promised, ℐ indexes a colimit which computes L. op × Construction 5.1.7. Note that ℐ admits a functor ℐ → (∆ ) sending (푛1)(푛2) ··· (푛푘) ∈ op × ℐ to {(푛1), (푛2), ··· , (푛푘)} ∈ (∆ ) . We associate to each simplicial space 푋∙ the functor X× : ℐ → 풮 given by the composite
푋× ℐ → (∆op)× −−→풮∙ ,
× op × where 푋∙ : (∆ ) → 풮 denotes the functor induced by 푋∙.
Proposition 5.1.8. Let 푋∙ be a simplicial space. Then the underlying space of the monoid L푋∙ is given by the formula
× L푋∙ ≃ colim X ≃ colim 푋푛1 × · · · × 푋푛푖 . ℐ (푛1)···(푛푖)∈ℐ
× op × Proof. Since B is restriction along the functor 퐹 : (∆ ) → 풯퐴, it suffices to show that left Kan extension along 퐹 × takes product preserving functors (∆op)× → 풮 to product preserving functors 풯퐴 → 풮, and that the Kan extension is given by the desired formula. We first compute the value of the left Kan extension at Z≥0 ∈ 풯퐴, which is op × indexed by the category 풦 := (∆ ) ×풯퐴 (풯퐴)/ Z≥0 . Unwinding the definitions, an object of 풦 is a pair (︀ )︀ {(푛1), (푛2), ··· , (푛푘)}, 푥
op × ∐︀ where {(푛1), (푛2), ··· , (푛푘)} ∈ (∆ ) and 푥 is a chosen element in Free( 푖(푛푖)). The morphisms in 풦 are maps in (∆op)× preserving this element. ∐︀ (푖) (푖) (푖) Notation. Inside Free( 푖(푛푖)), we will denote by 푥1 , 푥2 , ··· , 푥푛푖 the generators corresponding to (푛푖).
67 We define a functor 퐺 : ℐ → 풦 as follows. On objects, it sends (푛1)(푛2) ··· (푛푘) ∈ op × ℐ to {(푛1), (푛2), ··· , (푛푘)} ∈ (∆ ) together with the element given by the prod- uct (푥(1)푥(1) ··· 푥(1))(푥(2)푥(2) ··· 푥(2)) ··· (푥(푘)푥(푘) ··· 푥(푘)). 1 2 푛1 1 2 푛2 1 2 푛푘 A morphism
(푛1) ··· (푛푘) → (푚1) ··· (푚푗) in ℐ determines a canonical morphism
{(푛1), ··· , (푛푘)} → {(푚1), ··· , (푚푗)} in (∆op)× and it preserves the chosen elements of the corresponding free monoids because the morphisms in ℐ are assumed to be order preserving.
Lemma 5.1.9. The functor 퐺 admits a left adjoint 퐻 : 풦 → ℐ and therefore 퐺 is cofinal.
Proof of lemma. We begin by defining 퐻. Consider an object
(︀ )︀ {(푛1), (푛2), ··· , (푛푘)}, 푥 ∈ 풦.
(푖) The element 푥 is given by some word in the generators 푥푗 . We call a convex (푖) (푖) (푖) substring of 푥 (i.e., set of adjacent letters) snug if it is of the form 푥푗 푥푗+1 ··· 푥푗+푚−1 for some positive integers (푖, 푗, 푚). A snug substring of 푥 is called maximal if it is not a proper substring of a larger snug substring. Consider the unique partition of 푥 into maximal snug substrings
(푥(푖1)푥(푖1) ··· 푥(푖1) )(푥(푖2)푥(푖2) ··· 푥(푖2) ) ··· (푥(푖푙)푥(푖푙) ··· 푥(푖푙) ). 푗1 푗1+1 푗1+푚1−1 푗2 푗2+1 푗2+푚2−1 푗푙 푗푙+1 푗푙+푚푙−1
ℐ We regard this partition as an object 푥 ∈ ℐ (which is isomorphic to (푚1)(푚2) ··· (푚푙) ∈ ℐ), and define 퐻 on objects by
(︀ )︀ ℐ 퐻 {(푛1), (푛2), ··· , (푛푘)}, 푥 = 푥 .
(2) (1) (1) (1) (3) (3) (3) Example. If 푥 = 푥1 푥2 푥3 푥4 푥2 푥1 푥2 , then the partition into maximal snug (2) (1) (1) (1) (3) (3) (3) substrings is 푥 = (푥1 )(푥2 푥3 푥4 )(푥2 )(푥1 푥2 ), and 퐻 would send this to the object (1)(3)(1)(2) ∈ ℐ.
68 We now describe the effect of 퐻 on morphisms. Consider a morphism
(︀ )︀ (︀ )︀ {(푛1), (푛2), ··· , (푛푠)}, 푥 → {(푚1), (푚2), ··· , (푚푡)}, 푦 in 풦, and suppose 푥 and 푦 are written as a union of maximal snug substrings as follows:
푥 = (푥(푖1)푥(푖1) ··· 푥(푖1) )(푥(푖2)푥(푖2) ··· 푥(푖2) ) ··· (푥(푖푙)푥(푖푙) ··· 푥(푖푙) ) 푗1 푗1+1 푗1+ℎ1−1 푗2 푗2+1 푗2+ℎ2−1 푗푙 푗푙+1 푗푙+ℎ푙−1
푦 = (푦(푒1)푦(푒1) ··· 푦(푒1) )(푦(푒2)푦(푒2) ··· 푦(푒2) ) ··· (푦(푒푟)푦(푒푟) ··· 푦(푒푟) ) 푔1 푔1+1 푔1+푓1−1 푔2 푔2+1 푔2+푓2−1 푔푟 푔푟+1 푔푟+푓푟−1 (∙) (∙) (where the generators 푦∙ are defined analogously to the generators 푥∙ ). Such a morphism is determined by a map of sets 훾 : {1, ··· , 푡} → {1, ··· , 푠} together op with maps (푛훾(훼)) → (푚훼) in ∆ for each 1 ≤ 훼 ≤ 푡 such that the resulting map ∐︀ ∐︀ 휃 : Free( 훼(푚훼)) → Free( 훽(푛훽)) sends 푦 to 푥. We define the corresponding map ℐ ℐ ℐ (푖) (푒) 푥 → 푦 by sending the element of 푥 corresponding to 푥푗 to the element 푦푔 of 푦ℐ that hits it under 휃. This refines the partitions because under 휃, each maximal snug substring of 푦 is sent to a (possibly empty, but not necessarily maximal) snug substring of 푥.
It suffices now to exhibit appropriate unit and counit transformations. Itis immediate that 퐻 ∘ 퐺 is naturally the identity, so we will define the unit transfor- mation. Consider the object ({(푛1), (푛2), ··· , (푛푘)}, 푥) ∈ 풦, where
푥 = (푥(푖1)푥(푖1) ··· 푥(푖1) )(푥(푖2)푥(푖2) ··· 푥(푖2) ) ··· (푥(푖푙)푥(푖푙) ··· 푥(푖푙) ) 푗1 푗1+1 푗1+푚1−1 푗2 푗2+1 푗2+푚2−1 푗푙 푗푙+1 푗푙+푚푙−1 is a partition into maximal snug substrings. The functor 퐺 ∘ 퐻 sends this to the (︀ )︀ object {(푚1), (푚2), ··· , (푚푙)}, 푦 ∈ 풦 where
푦 = (푦(1)푦(1) ··· 푦(1))(푦(2)푦(2) ··· 푦(2)) ··· (푦(푙)푦(푙) ··· 푦(푙) ) 1 2 푚1 1 2 푚2 1 2 푚푙
(∙) (푦∙ as above). We then define the unit natural transformation on this object, which is the data of a map
(︀ )︀ (︀ )︀ {(푛1), (푛2), ··· , (푛푠)}, 푥 → {(푚1), (푚2), ··· , (푚푡)}, 푦
in 풦, by sending 푟 ∈ {1, ··· , 푡} to 푖푟 ∈ {1, ··· , 푠} and mapping (푛푖푟 ) to (푚푟) in the (푟) (푟) (푖푟) (푖푟) unique way such that the generators 푦1 , ··· , 푦푚푟 are sent to 푥푗푟 , ··· , 푥푗푟+푚푟−1,
69 respectively. It is easy to check that the triangle identities are satisfied and thus we have produced the adjoint.
This lemma implies that the value of the left Kan extension on Z≥0 is given by the desired formula. It suffices to show that the resulting Kan extended functor is product preserving. However, the argument above generalizes in a straightforward 푆 op × way to show that for any finite set 푆, there is a cofinal functor ℐ → (∆ ) ×풯퐴 (풯퐴)/Free(푆). This implies the Kan extended functor is product preserving because colim(푋푛 × 푋푛 × · · · × 푋푛 ) × (푋푚 × 푋푚 × · · · × 푋푚 ) ℐ×ℐ 1 2 푗 1 2 푘
≃ (colim 푋푛 × 푋푛 × · · · × 푋푛 ) × (colim 푋푚 × 푋푚 × · · · × 푋푚 ) ℐ 1 2 푗 ℐ 1 2 푘 where ((푛1)(푛2) ··· (푛푗), (푚1)(푚2) ··· (푚푘)) ∈ ℐ × ℐ, where we have used that colimits are universal in spaces.
5.2 The partial 퐾-theory of F푝
We now turn to the proof of Theorem 5.0.2.
Notation. 푆 = 푆 (Vectfd ) In the course of this proof, we set ∙ ∙ F푝 . All vector spaces will implicitly be over F푝.
The proof will proceed by applying the formula of Proposition 5.1.8 in the case 푋∙ = 푆∙. We aim to compute the colimit
colim 푆푛1 × 푆푛2 × · · · × 푆푛푗 (5.1) (푛1)···(푛푗 )∈ℐ of the functor S× : ℐ → 풮 (cf. Construction 5.1.7). Let 풵 → 풮 denote the universal left fibration, and let 풵0 → 풮 denote the left fibration classifying the functor 휋0 : 풮 → 풮. The natural transformation id풮 → 휋0 induces a map 푝 : 풵 → 풵0 of fibrations. Define ∞-categories 풳 and ℱ and functors 퐺 and 퐻 so that the squares in the following diagram are Cartesian:
풳 퐺 ℱ 퐻 ℐ
S× 푝 풵 풵0 풮 .
70 The composite 퐻 ∘ 퐺 is a left fibration because it arises as the pullback of the universal left fibration. It follows from [29, Corollary 3.3.4.6] that the colimit of S× is homotopy equivalent to |풳 |. On the other hand, the map 푝 is also a left fibration; explicitly, an object of 풵0 is a space 푋 together with an element 푎 ∈ 휋0(푋), and 푝 classifies the functor that sends this object to the component of 푋 containing 푎. It follows that 퐺 is itself a left fibration. Let S : ℱ → 풮 be the functor that classifies 퐺 (the reason for this notation will soon be clear). Applying [29, Corollary 3.3.4.6] again, we see that the colimit of S is also homotopy equivalent to |풳 |, and so we are reduced to computing the colimit of S. We turn to analyzing this colimit, starting with obtaining a more explicit description of ℱ.
Definition 5.2.1. Define a filtered dimension to be a nonempty sequence d = ⟨푑1, 푑2, ··· , 푑푘⟩ of nonnegative integers. We will soon think of d as keeping track of the dimensions of the successive quotients in the isomorphism class of filtered vector spaces determined by
푑1 푑1+푑2 푑1+···+푑푘 * ⊂ F푝 ⊂ F푝 ⊂ · · · ⊂ F푝 .
As such, we define its length to be 푙(d) = 푘 and its dimension to be |d| = 푑1 + 푑2 + ··· + 푑푘.
By definition, an object of ℱ is an object (푛1)(푛2) ··· (푛푘) ∈ ℐ together with a choice of element of 휋0(푆푛1 × 푆푛2 × · · · 푆푛푘 ). Unwinding the definitions, we see that this is the data of a (possibly empty) sequence 퐷 = (d(1), d(2), ··· , d(푗)) of filtered dimensions; we call such a sequence a filtered dimension sequence and define (푖) (푖) 푙(퐷) := Σ푖푙(d ) and |퐷| := Σ푖|d |. Explicitly, this correspondence between filtered dimension sequences and objects of ℱ sends the filtered dimension sequence (1) (2) (푗) 퐷 to (푙(d ))(푙(d )) ··· (푙(d )) ∈ ℐ together with the unique point in 휋0(푆푙(d(1))×
· · · × 푆푙(d(푗))) whose image in 휋0(푆푙(d(푖))) is the isomorphism class of filtered vector space determined by d(푖) for all 푖. We can then express the functor S : ℱ → 풮 classifying the fibration 퐺 by the formula
S(퐷) = 푆푙(d(1)) × · · · × 푆푙(d(푗)).
Next, we simplify the calculation of colim S by extracting a cofinal subcategory ℱ of ℱ:
71 Definition 5.2.2. Let ℱ red ⊂ ℱ be the full subcategory of filtered dimension se- quences which are reduced in the sense that each filtered dimension d = ⟨푑1, 푑2, ··· , 푑푘⟩ in the sequence satisfies 푑1, 푑2, ··· , 푑푘 ≥ 1.
Lemma 5.2.3. The inclusion ℱ red ⊂ ℱ admits a left adjoint, and is therefore cofinal.
Proof. The left adjoint is given by the functor red : ℱ → ℱ red which takes a filtered dimension sequence and removes all zeros (and any resulting empty filtered dimensions).
We are reduced to computing the colimit of the functor S : ℱ red → 풮 which, we remind the reader, has the following two properties:
1. A filtered dimension sequence consisting of a single filtered dimension d = ⟨푑1, 푑2, ··· , 푑푘⟩ is sent to the groupoid S((d)) of filtered vector spaces with filtered dimension d.
2. The functor S takes concatenation of filtered dimension sequences to prod- ucts of spaces; in other words, for 퐷 = (d(1), d(2), ··· , d(푗)) ∈ ℱ red, ∏︁ S(퐷) = S((d(푖))). 푖
Observation 5.2.4. The category ℱ red is a poset. The maps in ℱ red are gen- erated under concatenation of filtered dimension sequences by the following two operations:
1. For a filtered dimension d of length 푘 and an integer 1 ≤ 푖 < 푘, there is a collapse map
⟨푑1, 푑2, ··· , 푑푘⟩ → ⟨푑1, ··· 푑푖−1, 푑푖 + 푑푖+1, 푑푖+2, ··· 푑푘⟩.
2. For a filtered dimension d of length 푘 and an integer 0 < 푖 < 푘, there is a splitting map
⟨푑1, 푑2, ··· , 푑푘⟩ → (⟨푑1, 푑2, ··· , 푑푖⟩, ⟨푑푖+1, 푑푖+2, ··· , 푑푘⟩).
72 Since these maps preserve the total dimension |퐷|, we may write ℱ red as a disjoint union of posets red ∐︁ red ℱ = ℱ푚 푚 red red where ℱ푚 ⊂ ℱ is the subset spanned by filtered dimension sequences 퐷 such that |퐷| = 푚. This induces an equivalence: ∐︁ colim S(퐷) = colim S(퐷) 퐷∈ℱ red 퐷∈ℱ red 푚 푚 such that the 푚th component corresponds to 푚 ∈ Z≥0 under the isomorphism part 휋 (퐾 ( )) ≃ colim red S(퐷) 0 F푝 Z≥0. It remains to show that each ℱ푚 has trivial F푝- homology above degree zero. We proceed by induction on 푚. The case 푚 = 0 is red trivial. When 푚 = 1, the category ℱ푚 has a single element (⟨1⟩) which is sent to 퐵퐺퐿1(F푝), which has vanishing F푝-homology. Now, fix 푚 ≥ 2 and assume the ′ red statement for all 푚 < 푚. We analyze the diagram S : ℱ푚 → 풮 in detail.
red Definition 5.2.5. Define the following subsets of the poset ℱ푚 :
+ red 1. Let 풞푚 ⊂ ℱ푚 be the subset of filtered dimension sequences of the form (d) for a single filtered dimension d.
red 2. Let ℰ 푚 := {퐷 ∈ ℱ푚 | 푙(퐷) > 1} be the complement of the object (⟨푚⟩) in red ℱ푚 .
+ + 3. Let 풞푚 := ℰ 푚 ∩ 풞푚 = {퐷 ∈ 풞푚 | 푙(퐷) > 1} be the complement of the object + (⟨푚⟩) in 풞푚.
red + Note that ℱ = ℰ 푚 ∪ 풞푚 and there are no maps between ℰ 푚 − 풞푚 and 풞+ − 풞 ℱ red = ℰ ∐︀ 풞+ 푚 푚; it follows that there is a pushout of simplicial sets 푚 푚 풞푚 푚 and so by [29, Proposition 4.4.2.2], we have a homotopy pushout
colim S(퐷) colim S(퐷) + 퐷∈풞푚 퐷∈풞푚
colim S(퐷) colim S(퐷). red 퐷∈ℰ푚 퐷∈ℱ푚
Theorem 5.0.2 now follows from the following two propositions.
73 Proposition 5.2.6. The map
colim S(퐷) → colim S(퐷) + 퐷∈풞푚 퐷∈풞푚
+ induces an isomorphism in F푝-homology. Equivalently, since (⟨푚⟩) ∈ 풞푚 is a final object, the natural map colim S(퐷) → S(⟨푚⟩) 퐷∈풞푚 induces an isomorphism on F푝-homology.
Proposition 5.2.7. The space colimℰ푚 S(퐷) has trivial F푝-homology in positive degrees.
Remark 5.2.8. Proposition 5.2.6 is proven for all 푚 independent of the induction and so it will be used freely in the proof of Proposition 5.2.7 below.
Proof of Proposition 5.2.6. Let 풯푚 denote the poset of nontrivial proper subspaces 푚 of the vector space F푝 and let 풟 denote the category of nondegenerate simplices of 풯푚. There is a canonical inclusion 풯푚 ˓→ 풟 which is equivariant for the natural action of 퐺 = 퐺퐿푚(F푝). It therefore induces a homotopy equivalence |풯푚 //퐺| ≃ |풟 //퐺|. On the other hand, note that an object of 풟 is a flag of strict inclusions of 푚 nontrivial proper subspaces of F푝 , so there is a natural map 풟 → 풞푚 which is 퐺- equivariant (for the trivial action on 풞푚). Moreover, the induced map 풟 //퐺 → 풞푚 is precisely the Grothendieck construction applied to the functor S|풞푚 : 풞푚 → 풮 whose colimit we aim to compute. Therefore, we have equivalences
colim S(퐷) ≃ |풟 //퐺| ≃ |풯푚 //퐺| ≃ |풯푚|ℎ퐺. 퐷∈풞푚
By the Solomon-Tits theorem [42], the 퐺-space |풯푚| is a wedge of (푚 − 2)- 푚−2 dimensional spheres. Moreover, its top cohomology 퐻 (|풯푚|, F푝) is the Steinberg representation of 퐺 over F푝, which is irreducible and projective and therefore has vanishing homology in all degrees [20]. It follows from the homotopy orbit spectral sequence that the natural map |풯푚|ℎ퐺 → (*)ℎ퐺 induces an equivalence in F푝-homology, and so the natural map
colim S(퐷) → S(⟨푚⟩) 퐷∈풞푚
74 induces an isomorphism in F푝-homology as desired.
Proof of Proposition 5.2.7. We start by defining a filtration of ℰ 푚.
red 1. Let 풜푖 ⊂ ℱ푚 be the subset which are sequences of filtered dimensions d = ⟨푑1, 푑2, ··· , 푑푘⟩ such that 푑1, 푑2, ··· , 푑푘 ≤ 푖.
′ 2. Let 풜푖 ⊂ 풜푖 be the subset which are sequences of filtered dimensions d = ⟨푑1, 푑2, ··· , 푑푘⟩ such that either 푘 = 1 and 푑1 = 푖 or 푘 ≥ 1 and 푑1, 푑2, ··· , 푑푘 ≤ 푖 − 1.
′ Note that there are also natural inclusions 훿푖 : 풜푖 ˓→ 풜푖+1 for 푖 ≥ 1 and that ′ for 푖 ≥ 푚, 풜푖 = 풜푖+1 = ℰ 푚. Hence, the sequence
′ ′ 풜1 ⊂ 풜1 ⊂ 풜2 ⊂ 풜2 ⊂ · · · ℰ 푚 provides an exhaustive and finite filtration of ℰ 푚. In the following two lemmas, we show that the F푝-homology of the colimit of the functor S is unchanged as we move up this filtration.
′ Lemma 5.2.9. For all 푖 ≥ 1, the inclusion 풜푖 ⊂ 풜푖 is cofinal. Thus, the natural map colim S(퐷) → colim S(퐷) ′ 퐷∈풜푖 퐷∈풜푖 is an equivalence.
Proof. This is an inclusion of posets, so cofinality amounts to seeing that any ′ element of 풜푖 maps to an element in 풜푖. This is clear by repeatedly applying the splitting maps (cf. Observation 5.2.4).
′ Lemma 5.2.10. For 푖 ≥ 1, let (훿푖)! : Fun(풜푖, 풮) → Fun(풜푖+1, 풮) denote left ′ Kan extension along the inclusion 훿푖 : 풜푖 ˓→ 풜푖+1. Then the natural map (훿 ) (S| )(퐷) → S| ′ (퐷) 퐷 ∈ 푖 ! 풜푖 풜푖+1 induces an isomorphism on F푝-homology for all ′ 풜푖+1. Thus, the natural map
colim S(퐷) → colim S(퐷) ′ 퐷∈풜푖 퐷∈풜푖+1 induces an equivalence in F푝-homology.
75 ′ Proof. Consider some 퐷 ∈ 풜푖+1 − 풜푖 which we may write in the form
퐷 = (d1, d2, ··· , d푗).
We need to show that the map
colim S(퐷′) → S(퐷) ′ 퐷 ∈(풜푖)/퐷 induces an equivalence in F푝-homology. Consider the subset of ℳ ⊂ (풜푖)/퐷 con- ′ ′ ′ ′ sisting of filtered dimension sequences 퐷 = (d1, d2, ··· , d푗) satisfying the follow- ing conditions for 1 ≤ 푟 ≤ 푗 (recall the notation from Definition 5.2.1):
′ 1. |d푟| = |d푟|.
′ 2. If 푙(d푟) > 1, then d푟 = d푟.
In particular, 퐷 must be obtained from 퐷′ via some sequence of collapse maps. It is easy to see that the inclusion of posets ℳ ⊂ (풜푖)/퐷 is cofinal. But the poset ℳ splits as a product
ℳ = 퐶1 × 퐶2 × · · · × 퐶푗 퐶 ⊂ ℱ red 퐶 = {(d )} 푙(d ) > 1 퐶 = 풞 of posets 푟 |d푟|, where 푟 푟 if 푟 and 푟 |d푟| (from Definition 5.2.5) if 푙(d푟) = 1. Thus, we have:
colim S(퐷′) ≃ colim S(퐷′) ′ ′ 퐷 ∈(풜푖)/퐷 퐷 ∈ℳ ≃ colim S(퐷′ ) × · · · × colim S(퐷′ ) ′ 1 ′ 푗 퐷1∈퐶1 퐷푗 ∈퐶푗
≃푝 S((d1)) × · · · × S((d푗)) ≃ S(퐷)
where ≃푝 denotes F푝-homology equivalence, and we have used Proposition 5.2.6 for this equivalence.
Combining Lemmas 5.2.9 and 5.2.10, we conclude that the natural map
colim S(퐷) → colim S(퐷) = colim S(퐷) ′ 퐷∈풜1 퐷∈풜푚 퐷∈ℰ푚
76 ′ induces an equivalence in F푝-homology. But 풜1 is a one element set contain- ing only the filtered dimension sequence (⟨1⟩, ··· , ⟨1⟩) (with 푚 ⟨1⟩’s). Since 푚 푚 S((⟨1⟩, ··· , ⟨1⟩)) = S((⟨1⟩)) ≃ 퐵퐺퐿1(F푝) has no F푝-homology in positive de- grees, we are done.
77 78 Chapter 6
The 푝-complete Frobenius and the Action of 퐵 Z≥0
In this chapter, we combine the work of the previous chapters to prove Theorem A, which appears as Theorem 6.2.1. Throughout, we fix a prime 푝 and restrict attention to Frobenius maps corresponding to elementary abelian 푝-groups.
Notation. Let 푄VectF푝 ⊂ 풬 be the full subcategory spanned by elementary abelian 푝-groups (i.e., finite dimensional F푝-vector spaces).
퐹 In §6.1, we introduce the subcategory CAlg푝 ⊂ CAlg of 퐹푝-stable E∞-rings, which are roughly defined to be the 푝-complete E∞-rings for which all the gen- eralized Frobenius maps (cf. §3.2) can be regarded as endomorphisms. In §6.2, we turn to the proof of Theorem A. The oplax action of 풬 on CAlg from Theo- 퐹 rem 3.2.7 restricts to an action of 푄VectF푝 on CAlg푝 . Essentially by definition, −1 this extends to an action of the ∞-category 푄VectF푝 [ℒ ], which is identified with part 퐵퐾 (F푝) by Corollary 4.2.12.1. To finish, we use the F푝-homology equivalence part part 퐾 (F푝) → Z≥0 of Theorem 5.0.2 to pass from an action of 퐵퐾 (F푝) to an action of 퐵 Z≥0. We conclude §6.2 by recording Theorem 6.2.2, which describes the action of Frobenius on global algebras via partial 퐾-theory. While we do not apply this theorem later in the paper, we feel that it may be of independent in- terest. Finally, in §6.3, we include a brief discussion of 푝-perfect E∞-rings, which are the 퐹푝-stable E∞-rings for which the Frobenius is an equivalence. In particu- 1 lar, we note that the ∞-category of 푝-perfect E∞-rings admits an action of 푆 by Frobenius (Corollary 6.3.6.1).
79 6.1 퐹푝-stable E∞-rings
Definition 6.1.1. We say that a spectrum 푋 is 퐹푝-stable if 푋 is 푝-complete and for every finite dimensional F푝-vector space 푉 , the canonical map (Construction 3.2.3) can푉 : 푋 → 푋휏푉
퐹 is an equivalence. Let CAlg푝 ⊂ CAlg denote the full subcategory of E∞-rings whose underlying spectrum is 퐹푝-stable.
Remark 6.1.2. If 푋 is 퐹푝-stable and 푉 푊 is a surjection of F푝-vector spaces, then by examining the commutative diagram
푋휏푊 can푉 can푊 푊
푉 푋 can 푋휏푉 ,
푉 we see that the generalized canonical map can푊 is also an equivalence.
Warning 6.1.3. A discrete F푝-algebra is generally not 퐹푝-stable when regarded as a spectrum. For example, the Eilenberg-Maclane spectrum F푝 is not 퐹푝-stable 푡퐶푝 because F푝 has nontrivial homotopy groups in every degree.
∧ ∧ 푡퐶푝 We saw in Example 1.2.1 that the canonical map 푆푝 → (푆푝 ) is an equiv- alence by Lin’s theorem. This admits the following extension, due to work of Adams, Gunawardena, and Miller on the Segal conjecture for elementary abelian 푝-groups:
Theorem 6.1.4 (Adams-Gunawardena-Miller [1]). Let 푉 be a finite dimensional F푝-vector space. Then the canonical map
푉 ∧ ∧ 휏푉 can : 푆푝 → (푆푝 ) is an equivalence.
Corollary 6.1.4.1. The 푝-complete sphere is 퐹푝-stable. Since the proper Tate construction commutes with finite colimits, it follows that any spectrum which is finite over the 푝-complete sphere is also 퐹푝-stable.
80 Another family of 퐹푝-stable spectra will be important for us in Chapter 7.
Example 6.1.5 ([30], Example 5.2.7). Let 푅 be a (discrete) perfect F푝-algebra. + ∧ Then there is an essentially unique flat E∞-algebra 푊 (푅) over 푆푝 with the fol- lowing properties:
1. 푊 +(푅) is 푝-complete.
∧ + 2. The map induced on 휋0 by the unit map 푆푝 → 푊 (푅) is the natural map Z푝 → 푊 (푅).
∧ 3. For any connective 푝-complete E∞-algebra 퐴 over 푆푝 , the canonical map
+ MapCAlg ∧ (푊 (푅), 퐴) → HomF푝 (푅, 휋0(퐴)/푝) 푆푝
is an equivalence. In particular, these spaces are discrete.
We will refer to 푊 +(푅) as the spherical Witt vectors of 푅.
Proposition 6.1.6. Let 푅 be a (discrete) perfect F푝-algebra. Then the spectrum + 푊 (푅) is 퐹푝-stable.
+ Note that 퐹푝-stability is a condition only on the underlying spectrum of 푊 (푅), ∧ which is the 푝-completion of a direct sum of copies of 푆푝 . When 푅 is finite dimen- 휏푉 sional as an F푝-vector space, the proposition is immediate because (−) commutes with finite colimits. The content of the proposition is that the 푝-completion of an ∧ arbitrary direct sum of copies of 푆푝 remains 퐹푝-stable. We will need the following lemma.
Lemma 6.1.7. Let 퐼 be a set and let 푉 be a finite dimensional F푝-vector space. Then the following statements hold:
1. The natural map 푡푉︀(∧ ︀) 푡푉 ∧ (푆푝 ) → 푆푝 훼∈퐼 훼∈퐼 is 푝-completion.
2. The natural map 푡푉 ∧︀(∧ ︀) 푡푉︀(∧ ︀) 푆푝 → ( 푆푝 푝 ) 훼∈퐼 훼∈퐼 is an equivalence.
81 Proof. By [34, Lemma I.2.9], the target of (1) and both spectra in (2) are 푝- complete. Therefore, it suffices to show that the two maps are equivalences after smashing with the Moore spectrum 푆/푝. This is clear for (2), so we address (1). Since (−)푡푉 commutes with limits of Postnikov towers by [34, Lemma I.2.6], we have that 푡푉 푡푉 lim(휏≤푛푆/푝) ≃ (푆/푝) . (6.1) 푛 However, we can make a stronger statement about the convergence of this inverse limit. Note that by the Segal conjecture, 푆푡푉 is a finite wedge sum of spectra of the ∞ 푡푉 form Σ+ 퐵퐺 for finite groups 퐺; thus, all the homotopy groups of (푆/푝) are finite. Moreover, for each 푛, 휏≤푛푆/푝 has a finite number of nonzero homotopy groups, each of which is finite; thus, by the Tate spectral sequence, each term in thelimit (6.1) also has finite homotopy groups. It follows from [34, Lemma III.1.8] thatthe 푡푉 푡푉 pro-system 휋푖(휏≤푛푆/푝) is pro-constant with value 휋푖(푆/푝) . This statement is compatible with infinite direct sums and the lemma follows.
Corollary 6.1.7.1. Let 퐼 be a set and let 푉 be a finite dimensional F푝-vector space. Then the following statements hold:
1. The natural map 휏푉︀(∧ ︀) 휏푉 ∧ (푆푝 ) → 푆푝 훼∈퐼 훼∈퐼 is 푝-completion.
2. The natural map 휏푉 ∧︀(∧ ︀) 휏푉︀(∧ ︀) 푆푝 → ( 푆푝 푝 ) 훼∈퐼 훼∈퐼 is an equivalence.
Proof. We recall from Remark 2.3.5 that the proper Tate construction for 푉 is the cofiber of a map 퐶 → (−)ℎ푉 where 퐶 is a finite colimit of functors of the form ℎ푊 ((−) )ℎ푉/푊 for subgroups 푊 ⊂ 푉 . Letting 퐹 denote the cofiber of the natural
82 map (−)ℎ푉 → 퐶, we consider the diagram of functors
ℎ푉 푡푉 (−)ℎ푉 (−) (−)
퐶 (−)ℎ푉 (−)휏푉
퐹 * Σ퐹
where the rows and columns are cofiber sequences. By the above remarks, 퐹 is a 푡푊 finite colimit of functors of the form ((−) )ℎ푉/푊 for proper subgroups 푊 ⊂ 푉 . It follows from Lemma 6.1.7 that the functors (−)푡푉 and Σ퐹 both have the property that they send the 푝-completion map
∧︀(∧ ∧ 푆푝 → ( 푆푝 푝 훼∈퐼 훼∈퐼 to an equivalence. Thus, (−)휏푉 also has this property and part (2) is proved. 휏푉︀(∧ ︀) To prove part (1), we first claim that 훼∈퐼 푆푝 is 푝-complete. We have ∧ 푡푉︀(∧ ︀) seen above that 훼∈퐼 푆푝 is 푝-complete, so it suffices to see that 퐹 ( 훼∈퐼 푆푝 ) is as well. For this, we show that
푡푊︀(∧ ︀) ︀( 푡푊︀(∧ ︀)︀) 푆푝 ℎ푉/푊 ≃ 푆푝 ⊗ 퐵(푉/푊 ) 훼∈퐼 훼∈퐼 is 푝-complete for any subgroup 푊 ⊂ 푉 . This amounts to checking that the inverse limit of the system
푡푊︀(∧ ︀) 푡푊 푝︀(∧ ︀) 푡푊 푝︀(∧ ︀) 푝 ··· −→ 푆푝 ⊗퐵(푉/푊 ) −→ 푆푝 ⊗퐵(푉/푊 ) −→ 푆푝 ⊗퐵(푉/푊 ) 훼∈퐼 훼∈퐼 훼∈퐼
푡푊︀(∧ ︀) is zero. Since 훼∈퐼 푆푝 is bounded below, each homotopy group of this inverse limit only depends on a finite skeleton of 퐵(푉/푊 ). Thus, the statement follows 푡푊︀(∧ ︀) because the smash product of the 푝-complete spectrum 훼∈퐼 푆푝 with any finite complex is certainly 푝-complete. To finish, we claim that the natural map of (1) is an equivalence after smashing
83 with the Moore spectrum 푆/푝. Similarly to part (2), it suffices to prove the anal- 푡푊 휏푉 ogous statement for each functor ((−) )ℎ푉/푊 in place of (−) . But this follows by Lemma 6.1.7 and the fact that homotopy orbits commutes with infinite direct sums.
Proof of Proposition 6.1.6. Choose a basis {푥훼}훼∈퐼 for 푅 as an F푝-vector space. + + ∧ This determines a map 훼∈퐼 푆푝 → 푊 (푅) which exhibits 푊 (푅) as the 푝- ∧ completion of 훼∈퐼 푆푝 . The proposition then follows by combining Theorem 6.1.4 with Corollary 6.1.7.1.
6.2 The action of 퐵 Z≥0 on 퐹푝-stable E∞-rings
We now prove our main theorem.
Theorem 6.2.1 (Theorem A). There is an action of 퐵 Z≥0 on the ∞-category 퐹 CAlg푝 of 퐹푝-stable E∞-rings for which 푛 ∈ Z≥0 acts by the natural transformation 휙푛 : id → id.
Proof. The oplax monoidal functor of Theorem 3.2.7 restricts to an oplax monoidal functor
푄VectF푝 → Fun(CAlg, CAlg).
휏푉 By definition, the functor (−) fixes any 퐹푝-stable E∞-ring, so we obtain an oplax monoidal functor 퐹 퐹 Θ푝 : 푄VectF푝 → Fun(CAlg푝 , CAlg푝 ).
In fact, for F푝-vector spaces 푈, 푉 and an E∞-ring 퐴, the oplax structure map 퐴휏푈⊕푉 → (퐴휏푈 )휏푉 fits into a square
푉 퐴 can 퐴휏푉
can푈⊕푉 (can푈 )휏푉 퐴휏푈⊕푉 (퐴휏푈 )휏푉 .
When 퐴 is 퐹푝-stable, the three labeled maps are equivalences, and thus the bottom map is an equivalence as well. It follows that the functor Θ푝 is (strong) monoidal, rather than just oplax monoidal. Moreover, by the definition of 퐹푝-stable, Θ푝 has
84 the property that the backward morphisms ℒ ⊂ Mor(푄VectF푝 ) are sent to equiv- alences. Thus, by [28, Proposition 4.1.7.4], Θ푝 extends to a functor of monoidal ∞-categories −1 퐹 퐹 푄VectF푝 [ℒ ] → Fun(CAlg푝 , CAlg푝 ). By Corollary 4.2.12.1 and Proposition 4.1.8, we have an equivalence of ∞- part −1 part part categories 퐵퐾 (F푝) ≃ 푄VectF푝 [ℒ ]. The natural map 퐾 (F푝) → 휋0(퐾 (F푝)) ≃ Z≥0 induces the diagram of monoidal ∞-categories given by the solid arrows:
part 퐹 퐹 퐵퐾 (F푝) Fun(CAlg푝 , CAlg푝 ).
퐵 Z≥0
It suffices to exhibit a monoidal functor filling in the dotted arrow. ByRemark 4.1.7, this is equivalent to providing a lift in the diagram of E2-spaces
part 퐾 ( ) End(id 퐹 ) F푝 CAlg푝
Z≥0
End(id 퐹 ) where CAlg푝 denotes the E2-monoidal space of endomorphisms of the iden- 퐹 tity functor on the ∞-category CAlg푝 . By Theorem 5.0.2, the vertical map part 퐾 (F푝) → Z≥0 induces an isomorphism in F푝-homology. We would like to use this to show that the restriction map
part HomAlg (풮)( ≥0, End(id 퐹 )) → HomAlg (풮)(퐾 ( 푝), End(id 퐹 )) E2 Z CAlg푝 E2 F CAlg푝 is an equivalence. Note that for E2-spaces 푋 and 푌 , the space of E2-maps 푋 → 푌 is computed as a limit of mapping spaces from products of copies of 푋 to 푌 . Since products of F푝-homology isomorphisms are F푝-homology isomorphisms, it suffices End(id 퐹 ) 푝 to show that CAlg푝 is a -complete space. End(id 퐹 ) Hom 퐹 (퐴, 퐵) But CAlg푝 can be written as a limit of spaces of the form CAlg푝 for 푝-complete E∞-rings 퐴, 퐵. These spaces, in turn, are each the limit of mapping
85 spaces between 푝-complete spectra, which are 푝-complete. Since the full subcate-
푝 End(id 퐹 ) gory of -complete spaces is closed under limits, we conclude that CAlg푝 is 푝-complete and the proof is complete.
One can also prove a variant of this theorem for general global algebras:
Theorem 6.2.2.
1. The ∞-category CAlgGlo of global algebras admits an action of the monoidal ∞-category 퐵퐾part(Z) for which an abelian group 퐺 acts by the Frobenius 휙퐺 : id → id.
Glo Glo 2. The full subcategory CAlg푝 ⊂ CAlg of global algebras with 푝-complete underlying E∞-ring admits an action of 퐵 Z≥0 for which 1 ∈ Z≥0 acts by the Frobenius (for the group 퐶푝).
Proof. To see the first part, restrict the action of 3.3.6 to the full subcategory of 풬 spanned by the abelian groups (cf. Remark 3.2.6). This inverts the left morphisms and thus, by Corollary 4.2.12.1, gives an action of 퐵퐾part(Z). The proof of the second part is exactly analogous to the proof of Theorem 6.2.1.
1 6.3 Perfect E∞-rings and the action of 푆
Definition 6.3.1. An E∞-ring 퐴 is 푝-perfect if 퐴 is 퐹푝-stable and the Frobenius map 휙 : 퐴 → 퐴푡퐶푝 is an equivalence. We will denote the ∞-category of 푝-perfect perf E∞-rings by CAlg푝 .
Proposition 6.3.2. Let 퐴 be a 푝-perfect E∞-ring. Then for all finite dimensional F푝-vector spaces 푉 , the Frobenius map for 푉 (Construction 3.2.1)
휙푉 : 퐴 → 퐴휏푉 is an equivalence.
×푛 ×푛 Proof. It suffices to show that 휙퐶푝 : 퐴 → 퐴퐶푝 is an equivalence for all 푛. ×푛 ×푛 Note that inside 푄VectF푝 , the span (* ← 퐶푝 → 퐶푝 ) is the 푛-fold sum of ×푛 ×푛 (* ← 퐶푝 → 퐶푝), and similarly (* ← * → 퐶푝 ) = (* ← * → 퐶푝) . It follows from
86 Theorem 3.2.7 that there is a commutative diagram
((퐴푡퐶푝 )···)푡퐶푝
휙∘푛 can∘푛
×푛 휏퐶푝 퐴 ×푛 퐴 ×푛 퐴. 휙퐶푝 can퐶푝
×푛 But all the outer arrows except for 휙퐶푝 are assumed to be equivalences, so that one is as well.
∧ Example 6.3.3. The 푝-complete sphere 푆푝 is 푝-perfect. This is because the ∧ Frobenius is a map of E∞-rings, and since 푆푝 is the initial 푝-complete E∞-ring, ∧ 푡퐶푝 the Frobenius must be the unit map for (푆푝 ) . By the Lin’s theorem (Example 1.2.1), this map is an equivalence.
perf Example 6.3.4. Let 퐴 ∈ CAlg푝 and 푋 be a finite space. Then the mapping 푋 spectrum 퐴 is also a 푝-perfect E∞-ring. First, it is an E∞-ring because 푋 is canonically a coalgebra in spaces. Since the proper Tate construction commutes 푋 with finite colimits and 푋 is finite, 퐴 is 퐹푝-stable. To see that it is additionally 푝- perfect, note that the resulting functor 퐴(−) : 풮op → CAlg takes colimits of spaces 푡퐶푝 to limits of E∞-rings and the functor (−) : CAlg → CAlg preserves finite limits so the general case follows from the case 푋 = *, which is true by assumption. In particular, for any finite space 푋 (or the 푝-completion thereof), the spherical ∧ 푋 cochains (푆푝 ) is a 푝-perfect E∞-ring.
Example 6.3.5. Let 푅 be a (discrete) perfect F푝-algebra. Then the E∞-ring + 푊 (푅) of spherical Witt vectors is 푝-perfect. Since it is 퐹푝-stable by Proposi- tion 6.1.6, it suffices to check that the Frobenius map is an equivalence. Since + 푊 (푅) is 퐹푝-stable, we may identify the Frobenius map with an endomorphism + + 휙˜ : 푊 (푅) → 푊 (푅) of E∞-rings. Such an endomorphism is determined on 휋0 by the defining properties of spherical Witt vectors. By comparing withthe + E∞-Frobenius map for 휋0(푊 (푅)), we see that 휙˜ is in fact the map induced by the Frobenius on 푅, and is therefore an equivalence because 푅 was assumed to be perfect.
퐹 perf Example 6.3.6. The inclusion CAlg푝 ⊂ CAlg푝 is proper. For instance, by
87 ∧ Corollary 6.1.4.1, a square zero extension of 푆푝 by any nonzero spectrum which is ∧ finite over 푆푝 is 퐹푝-stable but not 푝-perfect.
The Frobenius action further simplifies when restricted to 푝-perfect algebras. By Theorem 6.2.1, we obtain a monoidal functor
perf perf 퐵 Z≥0 → Fun(CAlg푝 , CAlg푝 ).
By the definition of 푝-perfect, this restricted functor has the property that every morphism in 퐵 Z≥0 is sent to an equivalence. It therefore factors through the 1 group completion map 퐵 Z≥0 → 퐵 Z ≃ 푆 , and so we have:
∘ perf Corollary 6.3.6.1 (Theorem A ). The ∞-category CAlg푝 of 푝-perfect E∞-rings admits an action of 푆1 whose monodromy induces the Frobenius automorphism on each object.
Remark 6.3.7. Corollary 6.3.6.1 can also be deduced directly from Theorem 3.2.7 using Quillen’s computation of 퐾(F푝) (Theorem 5.0.1), and thus does not logically depend on the results of chapters 4 and 5. It does not, however, imply Theorem 6.2.1, which applies to the larger class of 퐹푝-stable E∞-rings.
88 Chapter 7
Integral Models for Spaces
In this final chapter, we apply our results on the Frobenius action to obtain models for spaces in terms of E∞-rings. We reiterate that the idea for this application is due to Thomas Nikolaus, who proposed it in the dual setting of 푝-complete coalgebras. In §7.1, we deduce Theorem B from Theorem A. Then, in §7.2, we deduce Theorem C from Theorem B.
7.1 푝-adic homotopy theory over the sphere
fsc Recall from the introduction that 풮푝 ⊂ 풮 denotes the full subcategory of spaces which are homotopy equivalent to the 푝-completion of a simply connected finite 휙=1 perf ℎ푆1 space and CAlg푝 := (CAlg푝 ) denotes the ∞-category of 푝-Frobenius fixed algebras. Here we prove:
fsc op ∧ 푋 Theorem B. (풮 ) → CAlg ∧ 푋 ↦→ (푆 ) The functor 푝 푆푝 given by 푝 lifts to a fully faithful functor ∧ (−) fsc op 휙=1 (푆푝 )휙=1 :(풮푝 ) → CAlg푝 .
The proof of Theorem B begins by noting that for any 푝-perfect E∞-ring 퐴, one can extract a 푝-Frobenius fixed algebra of “Frobenius fixed points” (Lemma + 푋 7.1.1). Composing this procedure with the functor 푋 ↦→ 푊 (F푝) (cf. Example ∧ (−) 6.1.5), we construct the functor (푆푝 )휙=1 in the statement of Theorem B (Con- struction 7.1.4). By construction, this latter functor is fully faithful if and only + (−) fsc op 푊 ( ) :(풮 ) → CAlg + if the functor F푝 푝 푊 (F푝) is fully faithful. We verify this by realizing an unpublished observation of Mandell which relates cochains with
89 + 푊 (F푝) coefficients to the previously understood case of F푝 cochains (Corollary 7.1.5.1).
Lemma 7.1.1. There is an adjunction
* perf ℎ푆1 perf 푖 : (CAlg푝 ) ↽−⇀− CAlg푝 : 푖* such that
• The left adjoint 푖* sends a 푝-Frobenius fixed algebra to its underlying 푝- perfect E∞-ring.
• Any 푝-perfect algebra 퐴 acquires an action of Z by Frobenius and the counit * ℎ Z map 푖 푖*퐴 → 퐴 is homotopic to the natural map 퐴 → 퐴.
ℎ Z Remark 7.1.2. The E∞-ring 퐴 is computed by the procedure
ℎ Z 퐴 ≃ lim 퐴 ≃ fib(id퐴 −휙퐴). 푆1
Thus, the above lemma says that given a 푝-perfect E∞-ring 퐴, one can take its “Frobenius fixed points” by taking a limit over its 푆1 orbit, and this procedure determines a 푝-Frobenius fixed algebra.
Proof. Let us consider 퐵푆1 as a pointed space, and denote the inclusion of the 푖 basepoint by {*} −→ 퐵푆1. The 푆1 action of Corollary 6.3.6.1 determines a pullback square perf CAlg푝 풟
푞 푓
{*} 푖 퐵푆1 where 푓 is a coCartesian (and Cartesian) fibration. Let sect∘(푓) ⊂ sect(푓) be the full subcategory of sections which send any morphism in 퐵푆1 to a coCartesian ∘ perf ℎ푆1 morphism in 풟. We have an equivalence sect (푓) ≃ (CAlg푝 ) , and the functor 푖* of the lemma statement is the restriction of a section of 푓 along 푖. The adjoint will be given by 푓-right Kan extension along 푖 [35, Definition 2.8]. perf By [35, Corollary 2.11], this exists as long as for any 퐴 ∈ CAlg푝 , the induced
90 diagram 1 1 푆 ≃ * ×퐵푆1 퐵푆*/ → 풟 has an 푓-limit. Since 푓 is a Cartesian fibration, this relative limit reduces to 1 perf perf taking a limit over a diagram 푆 → CAlg푝 , which exists because CAlg푝 has finite limits. Moreover, the limit is computed as the fiberof 1−휙 and the resulting section of 푓 is in the full subcategory sect∘(푓).
∧ + Notation. By analogy to the equivalence 푆푝 ≃ 푊 (F푝), we will hereafter use the ∧ + more compact notation 푆푝 := 푊 (F푝) to denote the spherical Witt vectors of F푝 (cf. Example 6.1.5).
∧ In Example 6.3.5, we saw that 푆 is a 푝-perfect ∞-ring and 휙 ∧ is the unique 푝 E 푆푝 ∧ ring endomorphism of 푆푝 which induces the 푝th power map on 휋0.
* ∧ ∧ Lemma 7.1.3. There is an equivalence of E∞-rings 푖 푖*푆푝 ≃ 푆푝 under which the * ∧ ∧ ∧ ∧ ∧ counit map 푖 푖*푆푝 → 푆푝 is homotopic to the unit map 푆푝 → 푆푝 for the ring 푆푝 .
∧ ∧ Proof. By Lemma 7.1.1, it suffices to see that the unit map 푆푝 → 푆푝 fits into a fiber sequence of spectra: 1−휙 ∧ ∧ ∧ 푆푝 ∧ 푆푝 → 푆푝 −−−−→ 푆푝 .
∧ The composite is zero because the Frobenius is the identity on 푆푝 . The conclusion follows by observing the isomorphism
∧ ∼ ∧ 휋*(푆푝 ) = 푊 (F푝) ⊗Z푝 휋*(푆푝 ) (7.1) and noting that the map induced by 1 − 휙 ∧ on homotopy groups is the map 푆푝 induced by 1 − 휙 : 푊 ( ) → 푊 ( ) 푊 (F푝) F푝 F푝 휙 on the first tensor factor of (7.1), where 푊 (F푝) is the usual Witt vector Frobenius.
Construction 7.1.4. We now construct the functor
∧ (−) fsc op perf ℎ푆1 (푆푝 )휙=1 :(풮푝 ) → (CAlg푝 )
∧ (−) op appearing in the statement of Theorem B. Consider the functor (푆푝 ) : 풮 →
91 CAlg . By Example 6.3.4, this functor takes 푝-completions of finite spaces to 푝- perfect E∞-rings, so it restricts to a functor
∧ (−) fsc op perf (푆푝 ) :(풮푝 ) → CAlg푝 .
Define the desired functor by the formula
∧ (−) ∧ (−) fsc op perf ℎ푆1 (푆푝 )휙=1 := 푖*(푆푝 ) :(풮푝 ) → (CAlg푝 ) .
* ∧ (−) ∧ (−) By Lemma 7.1.3, the functor 푖 (푖*(푆푝 ) ) agrees with (푆푝 ) as required by the theorem statement. The proof of Theorem B requires one additional fact.
Proposition 7.1.5 ([27] Theorem 3.1.9). Let 푋 be a space. Then, the E∞-algebra 푋 퐿 푋 F푝 is formally étale over F푝, i.e. the cotangent complex F푝 / F푝 is contractible. fsc Corollary 7.1.5.1. Let 푋 ∈ 풮푝 and let 푌 ∈ 풮 be any space. Then the natural map ∧ 푋 ∧ 푌 푋 푌 CAlg ∧ ((푆 ) , (푆 ) ) → CAlg ( , ) 푆푝 푝 푝 F푝 F푝 F푝 is an equivalence. Proof. Both sides send colimits in 푌 to limits in mapping spaces, so it suffices to consider the case 푌 = *. Since 푋 is assumed to be finite, the natural map
∧ 푋 푋 (푆 ) ⊗ ∧ 푝 → 푝 푝 푆푝 F F is an equivalence, and so there is an equivalence
푋 ∧ 푋 CAlg ( , 푝) ≃ CAlg ∧ ((푆 ) , 푝). F푝 F푝 F 푆푝 푝 F
Thus, it suffices to see that the natural map
∧ 푋 ∧ ∧ 푋 CAlg ∧ ((푆 ) , 푆 ) → CAlg ∧ ((푆 ) , 푝) 푆푝 푝 푝 푆푝 푝 F is an equivalence. This follows from the following two claims: Claim 1. For each 푛, the natural map
∧ 푋 ∧ 푋 CAlg ∧ ((푆 ) , 푊푛( 푝)) → CAlg ∧ ((푆 ) , 푝) 푆푝 푝 F 푆푝 푝 F
92 is an equivalence.
Claim 2. The natural map
∧ 푋 ∧ ∧ 푋 CAlg ∧ ((푆 ) , 푆 ) → CAlg ∧ ((푆 ) , 푊 ( 푝)) 푆푝 푝 푝 푆푝 푝 F is an equivalence.
Proof of Claim 1. The proof is by induction. The base case 푛 = 1 is a tautology. Assume the statement has been proven for 푛 and consider the map
∧ 푋 ∧ 푋 훾* : CAlg ∧ ((푆 ) , 푊푛+1( 푝)) → CAlg ∧ ((푆 ) , 푊푛( 푝)) 푆푝 푝 F 푆푝 푝 F induced by the projection 훾 : 푊푛+1(F푝) → 푊푛(F푝). ∧ 푋 We first note that the fiber of 훾* over any map 푥 ∈ CAlg ∧ ((푆 ) , 푊푛( 푝)) is 푆푝 푝 F nonempty: by Mandell’s theorem (Theorem 1.0.2) and the inductive hypothesis, the composite
∧ 푋 ∧ ∧ 푋 푋 → CAlg ∧ ((푆 ) , 푆 ) → CAlg ∧ ((푆 ) , 푊푛+1( 푝)) 푆푝 푝 푝 푆푝 푝 F 훾 ∧ 푋 ∧ 푋 −→ CAlg ∧ ((푆 ) , 푊푛( 푝)) ≃ CAlg ∧ ((푆 ) , 푝) ≃ 푋 푆푝 푝 F 푆푝 푝 F
∧ 푋 is an equivalence and thus exhibits the mapping space CAlg ∧ ((푆 ) , 푊푛( 푝)) as 푆푝 푝 F ∧ 푋 a retract of CAlg ∧ ((푆 ) , 푊푛+1( 푝)). But 훾 : 푊푛+1( 푝) → 푊푛( 푝) is a square 푆푝 푝 F F F zero extension in the sense of [28, §7.4.1]. Thus, by [28, Remark 7.4.1.8], it follows that the fiber of 훾* over the chosen map 푥 is a torsor for Mod ∧ 푋 (퐿 ∧ 푋 ∧ , 푝). (푆푝 ) (푆푝 ) /푆푝 F We finish by showing this space is contractible. Since 푋 is finite, we have an equivalence ∧ 푋 푋 (푆 ) ⊗ ∧ 푝 ≃ 푝 푆푝 F F푝 of F푝-algebras. By the base change properties of the cotangent complex, this yields an equivalence
푋 Mod ∧ 푋 (퐿 ∧ 푋 ∧ , F푝) ≃ Mod( )푋 (퐿 ∧ 푋 ∧ ⊗ ∧ 푋 F푝 , F푝) ≃ Mod 푋 (퐿 푋 , F푝). (푆푝 ) (푆푝 ) /푆푝 F푝 (푆푝 ) /푆푝 (푆푝 ) F푝 F푝 /F푝
This space is contractible because
푋 퐿 푋 ≃ F푝 ⊗ 푋 퐿 푋 / ≃ 0 F푝 /F푝 F푝 F푝 F푝
93 by Proposition 7.1.5.
∧ The proof of Claim 2 is similar, as 푆푝 → 푊 (F푝) factors as a sequence of square zero extensions by shifted copies of F푝. It follows that the composite
∧ 푋 ∧ ∧ 푋 ∧ 푋 CAlg ∧ ((푆 ) , 푆 ) → CAlg ∧ ((푆 ) , 푊 ( 푝)) → CAlg ∧ ((푆 ) , 푝) 푆푝 푝 푝 푆푝 푝 F 푆푝 푝 F is an equivalence, which concludes the proof of Corollary 7.1.5.1.
Theorem B is now immediate:
fsc Proof of Theorem B. Let 푋, 푌 ∈ 풮푝 . We would like to show that the natural map fsc 휙=1 ∧ 푋 ∧ 푌 풮푝 (푌, 푋) → CAlg푝 ((푆푝 )휙=1, (푆푝 )휙=1) is an equivalence. Unwinding the definitions, we have
휙=1 ∧ 푋 ∧ 푌 휙=1 ∧ 푋 ∧ 푌 CAlg푝 ((푆푝 )휙=1, (푆푝 )휙=1) ≃ CAlg푝 (푖*(푆푝 ) , 푖*(푆푝 ) ) * ∧ 푋 ∧ 푌 ≃ CAlg ∧ (푖 푖 (푆 ) , (푆 ) ) 푆푝 * 푝 푝 ∧ 푋 ∧ 푌 ≃ CAlg ∧ ((푆 ) , (푆 ) ) 푆푝 푝 푝 ∧ 푋 ∧ 푌 ≃ CAlg ∧ ((푆 ) , (푆 ) ) 푆푝 푝 푝 푋 푌 ≃ CAlg ( , )( ) F푝 F푝 F푝 Corollary 7.1.5.1 fsc ≃ 풮푝 (푌, 푋), (Theorem 1.0.2)
as desired.
7.2 Integral models for spaces
Here, we assemble Sullivan’s rational homotopy theory with Theorem B to prove (see Definition 1.2.6 for notation):
Theorem C. The functor (풮fsc)op → CAlg given by 푋 ↦→ 푆푋 lifts to a fully faithful functor (−) fsc op 휙=1 푆휙=1 :(풮 ) → CAlg .
94 (−) Proof. We start by constructing the functor 푆휙=1. For each prime 푝, the functor
∧ (−) fsc op perf (푆푝 ) :(풮 ) → CAlg푝 factors as a composite
(−) ∧ fsc op 푆 perf (−)푝 perf (풮 ) −−→ CAlg −−−→ CAlg푝 .
On the other hand, by Theorem B, the same functor also admits a factorization
(−) ∧ (푆∧) * fsc op (−)푝 fsc op 푝 휙=1 휙=1 푖 perf (풮 ) −−−→ (풮푝 ) −−−−−→ CAlg푝 −→ CAlg푝 .
By Definition 1.2.6, this determines a functor
(−) fsc op 휙=1 푆휙=1 :(풮 ) → CAlg lifting the functor 푆(−) :(풮fsc)op → CAlg.
We now show that this functor is fully faithful. As in the proof of Theorem B, it suffices to show that for any 푋 ∈ 풮fsc, the natural map
휙=1 푋 푋 → CAlg (푆휙=1, 푆휙=1) is an equivalence. This amounts to showing that the top square in the following diagram is a pullback square:
∏︀ 휙=1 ∧ 푋 ∧ 푋 푝 CAlg푝 ((푆푝 )휙=1, (푆푝 )휙=1)
푋 ∏︀ ∧ 푋 ∧ CAlg(푆 , 푆) 푝 CAlg((푆푝 ) , 푆푝 ) (7.2)
CAlg (푆푋 , 푆 ) ∏︀ CAlg ((푆∧)푋 , (푆∧) ). Q Q Q 푝 Q푝 푝 Q 푝 Q
Note that the bottom square is a pullback: this follows from considering maps
95 푋 of E∞-rings from 푆 into the pullback square
∏︀ ∧ 푆 푝 푆푝
∏︀ ∧ 푆Q 푝(푆푝 )Q.
It therefore suffices to show that the large rectangle in (7.2) is a pullback square.
Lemma 7.2.1. Let 푌 be a simply connected 푝-complete space such that 휋*(푌 ) is a finitely generated Z푝-module in each degree. Then the natural map
푌 → CAlg ( 푌 , ) Q푝 Q푝 Q푝
CAlg ( 푌 , ) 푌 exhibits Q푝 Q푝 Q푝 as the rationalization of .
Proof. This follows directly from the methods of [27, Proposition 1.3.3].
Let 푋 be a finite simply connected space as above. Then, since the functor 푌 ∧ 푌 ↦→ Q푝 sends the 푝-completion map 푋 → 푋푝 to an equivalence, we conclude that 푋 → CAlg ( 푋 , ) 푝 the natural map Q푝 Q푝 Q푝 is -completion followed by rationalization. Combining this with Sullivan’s rational homotopy theory (Theorem 1.0.1) and Theorem B, we see that the large rectangle in (7.2) is equivalent to the square
∏︀ ∧ 푋 푝 푋푝
∏︀ ∧ 푋Q 푝(푋푝 )Q.
This is a pullback square by work of Sullivan [44, Proposition 3.20], so the proof is complete.
7.3 Further extensions and questions
We include here a series of remarks highlighting additional questions which are not addressed in this paper. 1. Mandell’s theorem (Theorem 1.0.2) applies to finite type spaces which are not necessarily finite. The author does not know if there is a version ofTheorem
96 B which works in this generality. For example, Mandell’s theorem applies to the space 퐵퐶푝, providing an identification
퐵퐶푝 HomCAlg ( , 푝) ≃ ℒ퐵퐶푝. F푝 F푝 F
On the other hand, by the Segal conjecture, one can see that the E∞-algebra ∧ 퐵퐶푝 (푆푝 ) fails to be 퐹푝-stable, so our results do not apply.
∧ 퐵퐶푝 ∧ Question 1. What is the space of E∞-ring maps HomCAlg((푆푝 ) , 푆푝 )?
2. Recall the 2-category 풬̃︀ from Remark 3.4.10, and let 푄^VectF푝 denote the full subcategory on groupoids of the form 퐵푉 where 푉 is an F푝-vector space. perf Our methods show that in fact the E∞-space |푄^VectF푝 | acts on CAlg푝 (and the analogous statement in the non-group-complete setting).
Question 2. What is the homotopy type of |푄^VectF푝 |? For instance, is it 푝- adically discrete? What about the corresponding partial 퐾-theory?
3. In §3, we defined a notion of global algebra which has genuine equivariant multiplication maps corresponding to all finite covers of groupoids. It is natu- ral to consider the analogous object with multiplicative transfers for all maps of groupoids:
Definition 7.3.1. An extended global algebra is a section of the fibration Ψ+ : + Glo+Sp → Glo+ which is coCartesian over the left morphisms. Let CAlgGlo denote the ∞-category of extended global algebras.
One can show that the functor 푆(−) :(풮fsc)op → CAlg refines to a functor (−) fsc op Glo+ 푆Glo+ :(풮 ) → CAlg . The additional functorialities corresponding to geo- metric fixed points can be seen as giving trivializations of the Frobenius maps.We therefore conjecture that a space can be recovered from its cochains as an extended global algebra:
(−) fsc op Glo+ Conjecture 7.3.2. The functor 푆Glo+ :(풮 ) → CAlg is fully faithful.
97 98 Appendix A
Generalities on Producing Actions
In this appendix, we will provide the technical details necessary for producing the integral Frobenius action of Theorem A♮ (Theorem 3.2.7). For the sake of motivation, let us first describe a simpler case of the setup. Let 푞 : ℰ → 푆 be a co- Cartesian fibration of ∞-categories. There is a monoidal ∞-category Fun(푆, 푆)/ id whose objects can be thought of as pairs (휙, 휂) where 휙 ∈ Fun(푆, 푆) and 휂 : 휙 → id is a natural transformation. The monoidal structure is described on objects by the formula 휂1 휂2 (휙1, 휂1) ∘ (휙2, 휂2) = (휙1 ∘ 휙2, 휙1 ∘ 휙2 −→ 휙2 −→ id).
In this situation, the sections of 푞 admit a natural action of the monoidal ∞- category Fun(푆, 푆)/ id. Concretely, the action of (휙, 휂) on a section 휎 : 푆 → ℰ produces a new section which takes 푠 to (휂푠)*휎(휙(푠)). We work with an extension of this situation: for the remainder of this appendix, fix a pullback square of ∞-categories
ℰ ℰ +
푞 푞+ 푆 푖 푆+
where the vertical arrows are coCartesian fibrations and 푖 is a subcategory. We will see that the extension of 푞 to 푞+ : ℰ + → 푆+ describes additional symmetries on the sections of 푞 beyond those coming from Fun(푆, 푆)/ id as above. In particular, we + produce an action of the ∞-category Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖. Its objects
99 are the data of a functor 휙 : 푆 → 푆 together with a natural transformation 휂 : 푖휙 → 푖. The monoidal structure can be described informally by the formula
휂1 휂2 (휙1, 휂1) ∘ (휙2, 휂2) = (휙1 ∘ 휙2, 푖휙1 ∘ 휙2 −→ 푖휙2 −→ 푖).
Before we construct the monoidal structure formally, we need some preliminary lemmas. Lemma A.0.1. Let 풞, 풟 be ∞-categories and
퐹 : 풞 ↽−⇀− 풟 : 퐺 be functors such that 퐹 is left adjoint to 퐺 and 퐺 is fully faithful (so 퐹 is a localization in the sense of [29, Section 5.2.7]). Then there is a natural oplax monoidal functor Fun(풞, 풞) → Fun(풟, 풟) which sends a functor 휙 : 풞 → 풞 to 퐹 휙퐺 : 풟 → 풟. Proof. The simplicial monoid Fun(풞, 풞) admits a sub-simplicial monoid Funloc(풞, 풞) defined as the full simplicial subset spanned by those functors 휙 with the prop- erty that 휙(푐) ∈ 풟 for all 푐 ∈ 풞. There are obvious maps of simplicial monoids 푖 : Funloc(풞, 풞) → Fun(풞, 풞) and 푗 : Funloc(풞, 풞) → Fun(풟, 풟) which induce corre- sponding monoidal functors. But Funloc(풞, 풞) can be identified with Fun(풞, 풟) so 푖 admits a left adjoint, which is then oplax monoidal by [28, Proposition 2.2.1.1]. Composing this left adjoint with 푗 yields the desired oplax monoidal functor Fun(풞, 풞) → Fun(풟, 풟). Remark A.0.2. In fact, the hypotheses of Lemma A.0.1 are stronger than should be necessary – it should be sufficient that 퐹 and 퐺 be functors together with a natural transformation id → 퐺퐹. We would be happy to see a proof of this stronger statement. Lemma A.0.3. Let 퐵 be an ∞-category and 푞 : ℰ → 퐵 be a Cartesian fibration. Then the total space ℰ acquires a natural action of the monoidal ∞-category End(퐵)/ id where the action of (휂 : 휑 → id) ∈ End(퐵)/ id takes (푏 ∈ 퐵, 푥 ∈ ℰ 푏) ∈ ℰ * to (휑(푏), 휂푥푥 ∈ ℰ 휑(푏)).
Proof. The monoidal ∞-category End(퐵)/ id can be modeled explicitly by a sim- plicial monoid (which we will also denote by End(퐵)/ id) characterized as follows:
100 for any finite nonempty linearly ordered set 퐽, let 퐽 + denote 퐽 ∪ {+} where {+} 퐽 is a new maximal element; then, the set of maps ∆ → End(퐵)/ id is given by the + + set of maps ∆퐽 × 퐵 → 퐵 with the property that the restriction to {+} ⊂ ∆퐽 + is the identity. The monoid structure takes two maps 푓, 푔 : ∆퐽 × 퐵 → 퐵 to the composite
+ + + id ×푔 + 푓 푔 ∘ 푓 : ∆퐽 × 퐵 → ∆퐽 × ∆퐽 × 퐵 −−−→ ∆퐽 × 퐵 −→ 퐵
+ where the first map is induced by the diagonal on ∆퐽 .
The simplicial monoid End(퐵)/ id left acts on the underlying simplicial set of the arrow category Arr(퐵) = Fun(∆1, 퐵) of 퐵. On objects, the action of (휂 : 휑 → id) ∈ End(퐵)/ id on an arrow (푏 → 푐) ∈ Arr(퐵) produces an arrow 휂 representing the composite 휑(푏) −→푏 푏 → 푐. + In general, we describe this action directly on 퐽-simplices; let 푓 : ∆퐽 ×퐵 → 퐵 퐽 1 determine a 퐽-simplex of End(퐵)/ id and 훾 : ∆ × ∆ → 퐵 be a 퐽-simplex of 퐽 1 퐽+ Arr(퐵). There is a map 휓퐽 : ∆ × ∆ → ∆ which sends (푗, 0) to 푗 and (푗, 1) to + for all 푗 ∈ 퐽. The action on 퐽-simplices produces from 푓 and 훾 a new 퐽-simplex of Arr(퐵) defined by the composite
휓 ×id + id ×훾 + 푓 푓훾 : ∆퐽 × ∆1 −−−→퐽 ∆퐽 × (∆퐽 × ∆1) −−−→ ∆퐽 × 퐵 −→ 퐵.
It is immediate that this defines a left action, and so we have an action at the level of ∞-categories of the monoidal ∞-category End(퐵)/ id on Arr(퐵). Moreover, the action fixes the target of the arrow; thus, if we consider Arr(퐵) as lying over 퐵 via the target map, End(퐵)/ id acts on Arr(퐵) as an object over 퐵. Thus, for any functor 푞 : ℰ → 퐵, the fiber product Arr(퐵) ×퐵 ℰ acquires a left action of End(퐵)/ id.
One has a fully faithful functor ℰ → Arr(퐵) ×퐵 ℰ given by sending 푥 ∈ ℰ to 푥 together with the identity arrow 푞(푥) → 푞(푥). When 푞 is a Cartesian fibration, this functor admits a right adjoint which takes 푥 ∈ ℰ together with an arrow 푓 : 푦 → 푞(푥) to the pullback 푓 *푥. By (the opposite of) Lemma A.0.1, we obtain a lax action of End(퐵)/ id on ℰ. Explicitly, the action of (휂 : 휑 → id) ∈ End(퐵)/ id * takes (푏 ∈ 퐵, 푥 ∈ ℰ 푏) ∈ ℰ to (휑(푏), 휂푥푥 ∈ ℰ 휑(푏)). We immediately see that the lax structure map is an equivalence, and so we have produced the desired action.
Our strategy for producing and understanding the monoidal structure on Fun(푆, 푆)×Fun(푆,푆+)
101 + Fun(푆, 푆 )/푖 is to construct an ∞-category ℳ which is tensored over Cat∞ and + such that Fun(푆, 푆)×Fun(푆,푆+) Fun(푆, 푆 )/푖 arises as the endomorphism ∞-category of an object of ℳ in the sense of [28, Section 4.7.1].
Construction A.0.4. Let ℳ → Cat∞ denote the Cartesian fibration classified op by the functor Cat∞ → Cat∞ sending an ∞-category 풞 to the functor category Fun(풞, 푆+)op. One thinks of the objects of ℳ as pairs (풞, 휙), where 풞 is an ∞- category together with a functor 휙 : 풞 → 푆+. By Lemma A.0.3, the ∞-category ℳ acquires an action of End(Cat∞)/ id. Since Cat∞ is a Cartesian monoidal ∞-category, there is a monoidal functor (Cat∞)/* → End(Cat∞)/ id where * de- notes the terminal ∞-category. Since * is terminal, the natural monoidal functor (Cat∞)/* → Cat∞ is an equivalence. We conclude that the ∞-category ℳ is nat- urally left tensored over Cat∞ in the sense of [28, Definition 4.2.1.19]; explicitly, for 풟 ∈ Cat∞, the tensor is described by the formula
휙 퐷 ⊗ (풞, 휙) = (풟 × 풞, 풟 × 풞 → 풞 −→ 푆+).
It therefore makes sense to discuss endomorphism ∞-categories of objects of 푀.
Lemma A.0.5. The object (푆, 푖) ∈ ℳ admits an endomorphism ∞-category + which is equivalent to Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖.
Proof. Unwinding the definitions, there is a natural map in ℳ:
+ ev : Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖 ⊗ (푆, 푖) → (푆, 푖).
+ We would like to check that this exhibits Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖 as an endomorphism object for (푆, 푖) ∈ ℳ in the sense that for any ∞-category 퐾, the map ev induces a homotopy equivalence of spaces
+ ∼ + HomCat∞ (퐾, Fun(푆, 푆) ×Fun(푆,푆 ) Fun(푆, 푆 )/푖) −→ Homℳ(퐾 ⊗ (푆, 푖), (푆, 푖)).
To verify this, we note that both sides admit compatible maps to HomCat∞ (퐾 × 푆, 푆). It suffices to choose a particular functor 휓 : 퐾 × 푆 → 푆 and check the claim fiberwise over 휓. On the left hand side, this fiber is the fiber of the natural map + + + Fun(퐾, Fun(푆, 푆 )/푖) → Fun(퐾, Fun(푆, 푆 )) over the map 푖 ∘ 휓 : 퐾 × 푆 → 푆 .
102 This can be identified as the space of natural transformations filling in thediagram
휓 퐾 × 푆 푆
푖 푆 푖 푆+ where the left vertical arrow is projection onto the second coordinate. This is evidently also the fiber on the right hand side and ev induces the desired equiva- lence.
+ By [28, Section 4.7.1], this endows Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖 with the structure of a monoidal ∞-category. Moreover, this description as an endomor- phism object allows us to construct actions of the monoidal ∞-category Fun(푆, 푆)×Fun(푆,푆+) + Fun(푆, 푆 )/푖 on other ∞-categories. We will construct the right action of Fun(푆, 푆)×Fun(푆,푆+) + Fun(푆, 푆 )/푖 on sect(푞) by first producing an action on a closely related ∞- category.
+ + + Definition A.0.6. Let sect (푞) denote the ∞-category Fun(푆, ℰ )×Fun(푆,푆+)Fun(푆, 푆 )/푖.
Observation A.0.7. The ∞-category sect+(푞) can be identified as the ∞-category of maps in ℳ from (푆, 푖) to (ℰ +, 푞+). The verification is identical to Lemma A.0.5 so we omit it. As a result, sect+(푞) admits a canonical right action of + Endℳ((푆, 푖)) ≃ Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖.
Remark A.0.8. Concretely, an object of sect+(푞) is a pair (푓, 휂) where 푓 : 푆 → ℰ + is a functor together with a natural transformation 휈 : 푞+ ∘ 푓 → 푖. The action of + ′ + (휙, 휂) ∈ Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖 on (푓, 휈) produces (푓 ∘ 휙, 휈 ) ∈ sect (푞) where 휈′ denotes the composite
휈휙(−) 휂 푞+푓휙 −−−→ 푖휙 −→ 푖.
+ Note that Fun(푆, 푆 )/푖 has a final object given by 푖 ≃ 푖. The inclusion of this + final object induces a fully faithful embedding {*} → Fun(푆, 푆 )/푖, from which
103 one obtains a fully faithful embedding 푗 via the diagram
푗 sect(푞) sect+(푞)
(A.1) + + + Fun(푆, ℰ ) ×Fun(푆,푆+) {*} Fun(푆, ℰ ) ×Fun(푆,푆+) Fun(푆, 푆 )/푖.
Proposition A.0.9. The inclusion 푗 : sect(푞) → sect+(푞) admits a left adjoint.
Proof. We will apply the theory of marked simplicial sets as developed in [29, Section 3.1] and freely use the notation therein. Let (푓, 휂) ∈ sect+(푞). We will start by defining a section 푓 ′ ∈ sect(푞) together with a map 휃 :(푓, 휂) → 푗(푓 ′) in sect+(푞). Concretely, 푓 ′ will be given by the ′ formula 푓 (푠) = (휂푠)*푓(푠) for 푠 ∈ 푆. Consider the diagram of marked simplicial sets
푆♭ × {0} (ℰ +)♮
푞+ 푆♭ × (∆1)♯ (푆+)♯ where the bottom arrow encodes the natural transformation 휂 : 푞+ ∘ 푓 → id and the top arrow is defined by 푓. The opposite of the left vertical arrow is marked anodyne, and 푞+ is a coCartesian fibration, so we obtain a lift 푆 ×∆1 → ℰ +, which determines a map 휃 from (푓, 휂) to its restriction to 푆 × {1}, which we define to be the desired section 푓 ′. To show that 푗 has an left adjoint, it suffices to check that for any section 휎 ∈ sect(푞), 휃 induces an equivalence
′ Homsect(푞)(푓 , 휎) ≃ Homsect+(푞)((푓, 휂), 푗(휎)).
We can compute these mapping spaces by expressing sect+(푞) and sect(푞) as pullbacks as in diagram (A.1). In what follows, we will abuse notation by regarding the section 휎 as a map 휎 : 푆 → ℰ + together with an identification of 푞+휎 with 푖, and similarly with 푓 ′. Unwinding the definitions, one needs to show that 휃 induces an equivalence between the following two spaces:
104 ′ + ′ + 1. The fiber of the map of spaces HomFun(푆,ℰ+)(푓 , 휎) → HomFun(푆,푆+)(푞 ∘푓 , 푞 ∘ 휎) over the canonical element (corresponding to the fact that 푓 ′ and 휎 are + ′ + sections) which we will call 휈0 : 푞 ∘ 푓 → 푞 ∘ 휎.
+ + 2. The fiber of the map of spaces HomFun(푆,ℰ+)(푓, 휎) → HomFun(푆,푆+)(푞 ∘푓, 푞 ∘ + + 휎) over the natural transformation 휈1 : 푞 ∘ 푓 → 푞 ∘ 휎 given by precompo- + + ′ sition of 휈0 with the natural transformation 휈2 : 푞 ∘ 푓 → 푞 ∘ 푓 determined by 휃.
2 + The natural transformations 휈0, 휈1, 휈2 determine a 2-simplex 휈 : ∆ → Fun(푆, 푆 ) 2 by sending the edge opposite 푘 to 휈푘. We will consider the following subsets of ∆ as marked simplicial sets where the edge [0, 1] is marked:
∐︀ 푖1 2 [0, 1] {2} Λ1
푖0 푗1 (A.2)
2 푗0 2 Λ0 ∆ .
We have a commutative diagram
[0, 1] ∐︀{2} Fun(푆, ℰ +)
+ 푞* (A.3) ∆2 Fun(푆, 푆+) where the top horizontal map sends [0, 1] to the map 푓 → 푓 ′ induced by 휃 and + sends {2} to 휎, and 푞* is a coCartesian fibration by [29, Proposition 3.1.2.1].
The space (1) above is the space of dotted lifts in the diagram
[0, 1] ∐︀{2} Fun(푆, ℰ +)
+ 푖1 푞* 2 + Λ1 Fun(푆, 푆 ).
105 Analogously, the space (2) is the space of dotted lifts in the diagram
[0, 1] ∐︀{2} Fun(푆, ℰ +)
+ 푖0 푞* 2 + Λ0 Fun(푆, 푆 ).
However, note that the opposites of the inclusions 푗0 and 푗1 (from Diagram (A.2) above) are marked anodyne. Hence, by [29, Proposition 3.1.3.3], both spaces are equivalent to the space of lifts in Diagram (A.3) and thus are equivalent, as desired.
Applying Lemma A.0.1 to the adjunction of Proposition A.0.9, we find that there is an oplax monoidal functor
Fun(sect+(푞), sect+(푞)) → Fun(sect(푞), sect(푞)).
Composing with the right action of Observation A.0.7, we obtain an oplax monoidal functor
+ op (Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖) → Fun(sect(푞), sect(푞)).
In fact, the oplax structure maps are equivalences, and so we have shown:
Proposition A.0.10. Let 푞+ : ℰ + → 푆+ be a coCartesian fibration of ∞- categories, let 푖 : 푆 → 푆+ be a subcategory, and let 푞 denote the restriction of 푞+ + along 푖. Then there is a natural right action of Fun(푆, 푆) ×Fun(푆,푆+) Fun(푆, 푆 )/푖 휂 + on sect(푞) which, for 휎 ∈ sect(푞) and (휙 −→ id) ∈ Fun(푆, 푆)×Fun(푆,푆+) Fun(푆, 푆 )/푖, can be described on objects by the formula
휂 휎(휙 −→ id) = (푠 ↦→ 휂푠*휎(휙(푠))) ∈ sect(푞).
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