arXiv:1807.06000v3 [math.CT] 2 Sep 2019 fCatalonia. of anTheorem Main show: we precisely More toposes. presheaf even yetbihn that establishing by r ipiilst o simplicial (or sets simplicial are [ Kapranov and Dyckerhoff by geometry Here eopstosi naporaewys st nuecoalgeb induce to as ma so CULF way appropriate of an that in is decompositions spaces decomposition between morphisms Th of coalgebras. induce spaces decomposition one particular, categories In in as just decompose, (co-associatively) to [ Tonks and combi in purposes for introduced were spaces Decomposition Introduction 1 fcategories of l,tepouto w eopsto pcsa ipiils simplicial as spaces in decomposition product two bit of a appear product may the it and ple, maps, CULF and spaces decomposition of sraiyse ob tenreo)actgr,c.Lemma cf. category, a of) nerve (the be to seen readily is ipiilstpoutsol ahrb osdrda ten spaces. a tion as considered be rather should product simplicial-set a ipiilst(hc sadcmoiinsae sntte not is space) decomposition a is (which set simplicial nal † ∗ pr rmtecagbacapc,nts uhi nw about known is much so not aspect, coalgebraic the from Apart h rsn otiuinavne h aeoia td of study categorical the advances contribution present The pvkwsspotdb FS rnsF95–4103 n F and FA9550–14–1–0031 grants AFOSR by supported was Spivak okwsspotdb rnsMM06849P(E/EE,U (AEI/FEDER, MTM2016-80439-P grants by supported was Kock tw( oo,namely topos, oo.Peiey h lc aeoyoe n decompositi any over category slice the Precisely, topos. a D eso httectgr fdcmoiinsae n UFma CULF and spaces decomposition of category the that show We ) stetitdarwctgr of category arrow twisted the is eopsto-pc lcsaetoposes are slices Decomposition-space Decomp 7 , 8 , (Theorem 9 ,adproe nhmlgclagba ersnainthe representation algebra, homological in purposes and ], Decomp tepoetosgnrlyfi ob UF iial,tet the Similarly, CULF. be to fail generally projections —the Decomp oci Kock Joachim ∕ 4.5 D slclyatps enn htalissie r toposes— are slices its all that meaning topos, a locally is ≃ ). Decomp ∞ Psh For gopis ihapoet htepessteability the expresses that property a with -groupoids) (tw D ∗ D eopsto pc,teei aua equivalence natural a is there space, decomposition a ∕ Abstract ) D . 4 → ∼ ,wocl hmunital them call who ], 1 D Psh bandb deiesubdivision—this edgewise by obtained , ai .Spivak I. David (tw( D )) . 2.1 a ascaiey compose. (associatively) can nspace on otncl-eae class nicely-behaved most e o rdc o decomposi- for product sor A9550–17–1–0058. )o pi n 2017-SGR-1725 and Spain of E) below. aoisb ávz Kock, Gálvez, by natorics t snttecategorical the not is ets mnlin rminal ahomomorphisms. ra eopsto spaces decomposition † 2 h category the Sglsae.They spaces. -Segal euir o exam- For peculiar. s hs preserve These ps. D sapresheaf a is si locally is ps Decomp r,and ory, Decomp The . ermi- (1) The functor in the direction displayed in Eq. (1) is simply applying the twisted arrow category construction. The functor going in the other direction is the surprise. We construct it by exploiting an interesting natural transformation between the category of elements of the nerve and the twisted arrow category of a decomposition space:

el ◦N Decomp⇓ Cat. tw

Its component D ∶ el(ND) → tw(D) at a decomposition space D sends an n-simplex of D to its long edge. (In the category case, this map goes back to Thomason’s notebooks [18, p.152]; it was exploited by Gálvez–Neumann–Tonks [10] to exhibit Baues–Wirsching cohomology as a special case of Gabriel–Zisman cohomology.) The crucial property of  is that it is a cartesian natural transformation. Our proof of the Main Theorem describes the inverse to twD as being essentially—modulo some ∗ technical translations involving elements, presheaves, and nerves—given by D. Theimportance ofthis resultresidesin making a hugebodyof work in topostheory available to study decomposition spaces, such as for example the ability to define new decomposition spaces from old by using the internal language. It also opens up interesting questions such as how the subobject classifier [15], isotropy group [6], etc. of the topos Decomp∕D relate to the combinatorial structure of the decomposition space D. The Main Theorem can be considered surprising in view of the well-known failure of such a result for categories. Lamarche (1996) had suggested that categories CULF over a fixed base category C form a topos, but Johnstone [11] found a counter-example, and corrected the statement by identifying the precise—though quite restrictive— conditions that C must satisfy. Independent proofs of this result (and related condi- tions) were provided by Bunge–Niefield [3] and Bunge–Fiore [2], who were motivated by CULF functors as a notion of duration of processes, as already considered by Law- vere [13]. The present work also grew out of interest in dynamical systems [17]. Our main theorem can be seen as a different realization of Lamarche’s insight, allowing more general domains, thus indicating a role of decomposition spaces in category theory. From this perspective, the point is that the natural setting for CULF functors are decomposition spaces rather than categories: a simplicial set CULF over a category is a decomposition space, but not always a category.

2 Preliminaries

Simplicial sets. Although decomposition spaces naturally pertain to the realm of ∞- categories and simplicial ∞-groupoids, we work in the present note with 1-categories and simplicial sets, both for simplicity and in order to situate decomposition spaces (actually just “decomposition sets”) in the setting of classical category theory and topos theory. All the results should generalize to ∞-categories (in the form of Segal spaces) and general decomposition spaces, and the proof ideas should also scale to this

2 context, although the precise form of the proofs does not: where presently we exploit objects-and-arrows arguments, more uniform simplicial arguments are required in the ∞-case. We leave that generalization open. Thus our setting is the category sSet = Psh(∆) of simplicial sets, i.e. functors ∆op → Set and their natural transformations. Small categories fully faithfully em- bed as simplicial sets via the nerve functor N ∶ Cat → sSet, and decomposition spaces are defined as certain more general simplicial sets, as we now recall.

Active and inert maps. The category ∆ has an active-inert factorization system: the active maps, written g ∶ [m] ⇥ [n], are those that preserve end-points, g(0) = 0 and g(m) = n; the inert maps, written f ∶ [m] ↣ [n], are those that are distance preserving, f(i+1) = f(i) + 1 for 0 ≤ i ≤ m − 1. The active maps are generated by the codegeneracy i i maps s ∶ Xn → Xn+1 (for all 0 ≤ i ≤ n) and the inner coface maps d ∶ Xn → Xn−1 (for all 0

Decomposition spaces. Active and inert maps in ∆ admit pushouts along each other, and the resulting maps are again active and inert. A decomposition space [7] is a simplicial set (or more generally a simplicial groupoid or ∞-groupoid) X ∶ ∆op → Set that takes all such active-inert pushouts to pullbacks:

⎛ ¨ ⎞ [n ] [n] Xn¨ Xn ⎜ ⌟ ⎟ ⌟ X ⎜ ⎟ = ⎜ ⎟ ⎜ ¨ ⎟ ⎝ [m ] [m] ⎠ Xm¨ Xm.

CULF maps. A simplicial map F ∶ Y → X between simplicial sets is called CULF [7] when it is cartesian on active maps (i.e. the naturality squares are pullbacks), or, equivalently, is right-orthogonal to all active maps Δ[m] ⇥ Δ[n]. The CULF maps between (nerves of) categories are precisely the discrete Conduché fibrations (see [11]). If D is a decomposition space (e.g. a category) and F ∶ E → D is CULF, then also E is a decomposition space (but not in general a category). We denote by Decomp the category of decomposition spaces and CULF maps.

Right fibrations and presheaves. A simplicial map is called a right fibration if it is cartesian on bottom coface maps (or equivalently, right-orthogonal to the class of last-vertex inclusion maps Δ[0] → Δ[n]). (This in fact implies that it is cartesian on all codegeneracy and coface maps except the top coface maps. In particular, a right fibration is CULF.) When restricted to categories, this notion coincides with that of discrete fibration. We denote by RFib the category of small categories and right C fibrations. Note that for any category , the inclusion functor RFib∕C → Cat∕C is full.

3 For any small category C there is an adjunction

C ) ∶ Cat∕C ⇆ Psh( ) ∶ with ⊢ ); the presheaf )(F ) associated to a functor F ∶ D → C is given by taking the left Kan extension )(F ) ≔ LanF (1) of the terminal presheaf 1 ∈ Psh(D) along F . The notation is chosen because )◦ ≅ id. For any X ∈ Psh(C), we denote X by C X ∶ el(X) → , and refer to el(X) as the category of elements; an object of el(X) is a pair (c, x) where c ∈ C and x ∈ X(c), and a morphism (c¨, x¨) → (c, x) is a morphism ¨ C ¨ f ∶ c → c in with X(f)(x) = x . The projection functor X is always a right C ¨ fibration, and ) restricts to an equivalence of categories RFib∕C ≃ Psh( ). If X → X is a map of presheaves, el(X¨) → el(X) is a right fibration. Thus we have a functor el∶ Psh(C) → RFib. We will make particular use of this for the case C = ∆:

el∶ sSet → RFib.

Twisted arrow categories. For C a small category, the twisted arrow category tw(C) (cf. [12]) is the category of elements of the Hom functor Cop × C → Set. It thus has the arrows of C as objects, and trapezoidal commutative diagrams ⋅ ⋅

f ¨ f ⋅ ⋅ as morphisms from f ¨ to f. The twisted arrow category is a special case of edgewise subdivision of a simplicial set [16], as we now recall. Consider the functor

Q∶ ∆ ⟶ ∆ [n] ⟼ [n]op ⋆ [n] = [2n+1].

With the following special notation for the elements of the ordinal [n]op ⋆ [n] = [2n+1], 0 1 ⋯ n (2) 0¨ 1¨ ⋯ n¨, the functor Q is described on arrows by sending a coface map di ∶ [n−1] → [n] to the monotone map that omits the elements i and i¨, and by sending a codegeneracy map si ∶ [n] → [n−1] to the monotone map that repeats both i and i¨. Defining sd ≔ Q∗ ∶ sSet → sSet, we have the commutative diagram

Cattw Cat N N sSet sSet. sd

4 Note that there is a natural transformation L∶ id∆ ⇒ Q, whose component at [n] is the last-segment inclusion [n] ⊆ [n]op ⋆ [n]. It induces a natural map

∗ codX ≔ L ∶ sd(X) → X for any simplicial set X, and similarly tw(C) → C for any category C.

Lemma 2.1. The functor sd∶ sSet → sSet sends decomposition spaces to categories and CULF maps to right fibrations. That is, there is a unique functor tw making the following diagram commute: Decomptw RFib

N N sSet sSet. sd

Note that the “nerve functor” N ∶ Decomp → sSet on the left of the diagram is just the inclusion, but it will be convenient to have it named, so as to stress that we regard decomposition spaces as structures generalizing categories.

Proof. Suppose D is a decomposition space; we need to check that in sd(ND), the inert face maps d⊤ and d⊥ form pullbacks against each other (the Segal condition). But each top face map in sd(ND) is given by the composite of two outer face maps in ND (removing n and n¨ in Eq. (2)), and each bottom face map in sd(ND) is given by the composite of two inner face maps in ND (removing 0 and 0¨). Hence the Segal condition on sd(ND) follows from the decomposition-space condition on ND. If E → D is CULF, its naturality square along any active map is cartesian. The bottom face maps of tw(E) are given by (composites of) inner—hence active—maps in E, and similarly for D, so the naturality square along any bottom face of tw(E) → tw(D) is again cartesian, as required for it to be a right fibration.

Remark 2.2. The object part of Lemma 2.1 has been observed also by Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer [1], who furthermore establish the following converse result (in the more general setting of simplicial objects X in a combinatorial model category): sd(X) is Segal if and only if X is a decomposition space. This interpretation of their result depends on the recent result from [5] that every 2-Segal space is unital.

3 From category of elements to twisted arrow category

In this section we describe the natural transformation from categories of elements to twisted arrow categories. First we shall need a few basic facts about Q and the “last-vertex map.”

Lemma 3.1. The functor Q sends top-preserving maps to active maps.

5 Proof. If f ∶ [m] → [n] preserves the top element (that is, f(m) = n), then Q(f) is the map f op ⋆f ∶ [m]op ⋆[m] → [n]op ⋆[n], which clearly preserves both the bottom element m¨ and the top element m.

The “last-vertex map” of [19] is a natural transformation last ∶ N◦ el ⇒ idsSet, which sends a simplex  in (N el X)0 to its last vertex. Its value in higher simplicial degree is given by the following lemma.

f f f ∆ 1 2 k Lemma 3.2. For any k, let ' ∈ (N )k denote a sequence of maps [n0] → [n1] → ⋯ → [nk] in ∆. Then there is a unique commutative diagram B(') of the form

⊤ ⊤ ⊤ [0]d [1] d ⋯ d [k]

(n0) (f1) (fk)

[n0] [n1] ⋯ [nk] f1 f2 fk for which all the vertical maps are top preserving, and all the maps in the top row are d⊤.

The proof is straightforward. For a hint, see the proof provided for the next lemma, where we give a two-sided refinement of this construction.

f f f ∆ 1 2 k Lemma 3.3. For any k, let ' ∈ (N )k denote a sequence of maps [n0] → [n1] → ⋯ → [nk] in ∆. Then there is a unique commutative diagram A(') of the form

Q(d⊤) Q(d⊤) Q(d⊤) Q[0] Q[1] ⋯ Q[k]

(n0) (f1) (fk) (3)

[n0] [n1] ⋯ [nk] f1 f2 fk i.e. for which all the vertical maps are active and all the top maps are of the form Q(d⊤).

Proof. When k = 0 it is clear: there is a unique active map Q[0] = [1] ⇥ [n0]; call it (n0). The result now follows from the obvious fact that for any map ℎ∶ Q[m] → [n] in ∆, the following solid arrow diagram admits a unique active extension as shown:

Q(d⊤) Q[m] Q[m+1]

ℎ [n]

Remark 3.4. In the k = 1 case of Lemma 3.3,

Q(d⊤) Q[0] Q[1]

(m) (f) [m] [n], f

6 the map (f)∶ [3] ⇥ [n] is described explicitly for any f ∶ [m] → [n] as follows:

0 ↦ 0, 1 ↦ f(0), 2 ↦ f(m), 3 ↦ n.

In particular, if f itself is active, so that f(0) = 0 and f(m) = n, then the map (f) is doubly degenerate, in the sense that it factors through Q(s0) as in the following diagram: Q(s0) [1] [3]

(m) (f) [m] [n]. f

Lemma 3.5. There is a natural transformation Λ∶ N◦ el ⇒ sd of functors sSet → sSet. The degree-0 component (N el X)0 → (sd X)0 sends any simplex in X to its long edge.

Proof. A k-simplex in N el(X) is a sequence [n0] → ⋯ → [nk] together with an nk-  ∈ X Q[k] ⇥ [n ] simplex nk . By Lemma 3.3, there is an induced map k , and hence Δ[k] → sd(X). Naturality follows from the uniqueness of the diagram in (3).

Remark 3.6. The two arguments in lemmas 3.2 and 3.3 can be compared by means of the last-segment inclusion Lm ∶ [m] → Q[m], by which cod was defined. One finds that Λ mediates between the natural transformations last and cod as follows:

N◦ elΛ sd

last cod id . ∆ Lemma 3.7. For any sequence ' ∈ (N )k as in lemmas 3.2 and 3.3, we have

Q(B(')) = A(Q(')).

Proof. Just observe that Q(B(')) is a diagram with bottom row Q('), and it satisfies the condition of Lemma 3.3, as a consequence of Lemma 3.2 and Lemma 3.1. Therefore, by uniqueness in Lemma 3.3, the diagram must be A(Q(')).

For X a simplicial set with corresponding right fibration p∶ el(X) → ∆, there is a functor !X ∶ el(sd X) ⇒ el(X) induced by pullback along Q:

! el(sd X)X el(X) ⌟ sd p p (4) ∆ ∆ Q

Itsendsan n-simplex in sd(X) to the corresponding (2n+1)-simplex in X. Thesefunctors assemble into a natural transformation

!∶ el ◦ sd ⇒ el .

7 Lemma 3.8. The following diagram in the category of endofunctors on sSet commutes:

N◦ el ◦ sd N! lastsd

N◦ el sd . Λ

Proof. Given an k-simplex x ∈ N el(sd X)k, i.e. a sequence [n0] → ⋯ → [nk] and morphism Δ[nk] → sd(X), applying lastsd returns the dotted arrow as shown (see Lemma 3.2):

Δ[d⊤] Δ[d⊤] Δ[d⊤] Δ[0] Δ[1] ⋯ Δ[k] lastsd X

∗ Δ[n0] Δ[n1] ⋯ Δ[nk] Q X.

Here we have written Q∗X instead of the usual sd(X) because we will make use of its adjoint Q!, which restricts to Q on representables, i.e. Q!Δ[n] = Δ[2n+1]. Applying instead Λ◦N! returns the dotted arrow as shown:

⊤ ⊤ ⊤ Q!Δ[d ] Q!Δ[d ] Q!Δ[d ] Q!Δ[0] Q!Δ[1] ⋯ Q!Δ[k] Λ◦N!(X)

Q!Δ[n0] Q!Δ[n1] ⋯ Q!Δ[nk] X.

To see that these two dotted maps represent the same k-simplex of sd(X) via the ∗ Q! ⊣ Q adjunction, we invoke Lemma 3.7 and the uniqueness from Lemma 3.3.

Proposition 3.9. On decomposition spaces, the map Λ restricts to a cartesian natural trans- formation ∶ el ◦N ⇒ tw: el ◦N Decomp⇓ Cat tw N N N◦ el sSet⇓Λ sSet sd

Proof. If D is a decomposition space, tw(D) is a category by Lemma 2.1, and so is el(ND). Because N ∶ Cat → sSet is fully faithful, the natural transformation Λ lifts uniquely to a natural transformation  as shown. It remains to show that for any CULF map F ∶ E → D, the diagram

 el(NE)E tw(E)

el(NF ) tw(F ) (5) el(ND) tw(D) D

8 is cartesian. On objects and morphisms respectively, this amounts to showing that a unique lift exists for any solid-arrow squares (arbitrary ,,f) as follows:

⊤ Q!Δ[d ]  Δ[1]  E Δ[1] Δ[3] E

(n) F (n0) (f) F n D Δ[n ] Δ[n ] D Δ[ ]  0 f 1 

Here the ’s denote the unique active maps, as in Lemma 3.3. Thesetwo lifts doindeed exist uniquely because F is CULF.

Remark 3.10. Let us spell out in explicit terms the functor D ∶ el(ND) → tw(D) for D a decomposition space. An object in el(ND) is a pair ([n],) with [n] ∈ ∆ and  ∶ Δ[n] → D. Now D([n],) is the long edge of , considered as an object in tw(D). An arrow in el(ND) from ([n¨],¨) to ([n],) is an arrow f ∶ [n¨] → [n] such that ¨ = f ∗. Such an arrow is sent by D to the 3-simplex D(f) = ◦ (f) in the diagram

Δ[1]

0 Q!Δ[d ] Q Δ[d1] Δ[1]! Δ[3]

D(f) (n¨) (f)

Δ[n¨] Δ[n] D , Δ[f]  where the square is as in Lemma 3.3 (and more specifically Remark 3.4). This D(f) 1 Q!Δ[d ] D(f) is regarded as an arrow in the category tw(D) with domain Δ[1] ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,→ Δ[3] ,,,,,,,,,,,,,,,,,,,,,,→ D 0 Q!Δ[d ] D(f) and codomain Δ[1] ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,→ Δ[3] ,,,,,,,,,,,,,,,,,,,,,,→ D, in the usual way. ¨ In this situation, if f ∶ [n ] ⇥ [n] is active, then D(f) is an identity arrow in tw(D), D 0 that is, a 3-simplex in of the form “Q!Δ[s ] of a 1-simplex”, as noted in Remark 3.4.

4 Proof of main theorem

Lemma 4.1. For every decomposition space D, the following diagram commutes (up to iso- morphism):

ND elND Decomp∕D sSet∕ND RFib∕ el(ND) Q∗ el(ND) (6) twD sdND RFib∕tw(D) sSet∕ sd(ND) RFib∕ el(sd ND) Ntw(D) elsd(ND)

∗ where Q ∶ RFib∕∆ → RFib∕∆ is pullback along Q.

9 Proof. We already have the following (up-to-iso) commutative diagram:

N el Decomp sSet RFib∕∆ Q∗ tw sd RFib sSet RFib ∆. N el ∕ Indeed, the left side side is Lemma 2.1, and the right side is just the translation between simplicial sets and right fibrations over ∆ given in Eq. (4). Now slice it over D. (As always when slicing (up-to-iso) commutative diagrams, the result involves the identifications already expressed by the diagrams. In the present case we use sd(ND) ≅ N tw(D) and Q∗(el ND) ≅ el(sd ND) ≅ el(N tw D).)

∗ A key ingredient in the proof of the main theorem is to see that D provides a sort of splitting of the double rhombus diagram from the previous lemma:

Lemma 4.2. For any D ∈ Decomp, the following diagram commutes up to natural isomor- phism.

ND elND Decomp∕D sSet∕ND RFib∕ el(ND) Q∗ el(ND) (7) ∗ twD D RFib∕tw(D) sSet∕ sd(ND) RFib∕ el(sd ND) Ntw(D) elsd(ND) Proof. The commutativity of the left square follows immediately from the fact that  is cartesian, cf. Proposition 3.9; see in particular the pullback square in Eq. (5). For the right square, let p∶ F → tw(D) be a right fibration. We must establish a NF Q∗ ∗ F ND map el( ) → el(ND)( D ) and show it is an isomorphism over el(sd ). ByEq.(4), Q∗ ! the functor el(ND) is pullback along ND in the following diagram of categories: F¨ F ⌟ ⋅ ⌟ p¨ p !  el(sd ND)ND el(ND)D tw D ⌟

∆ ∆ Q F¨ Q∗ ∗ F Thus we have ≅ el(ND)( D ). The nerve of the middle composite,

N! D ΛN N el(sd ND) ,,,,,,,,,,,,,,,,,,,,,,,,,,→N N el(ND) ,,,,,,,,,,,,,,,→ sd(ND), is identified with lastsd(ND) by Lemma 3.8 and Proposition 3.9, which in particular gives N ≅ΛN. The naturality square for the last-vertex map

last F N el(NF) N NF

N el(Np) Np N el(sd ND) sd(ND) lastsd(ND)

10 induces a morphism N el(NF) → NF¨ by the universality of F¨ as a pullback, and the fact that N ∶ Cat → sSet preserves limits. Since N is fully faithful, we obtain our desired comparison map u∶ el(NF) → F¨ of discrete fibrations over el(sd ND). Itis enough to check that the restriction of u to each fiber is a bijection; to do so we describe these fibers and the map u between them in concrete terms. An object in el(sd ND) is a pair ([n],) where  ∶ Δ[n] → sd(ND). Asan n-simplex in tw(D), itissentby !ND to the same pair ([n],), butwherenow  is considereda (2n+1)- simplex in D. Applying D returns the long edge of the simplex, Δ[1] ⇥ Δ[2n+1] → D, which we denote l ∈ Obtw(D). The fiber of F¨ over  is the p-fiber over l, that is the discrete set {z ∈ Ob F ∣ p(z) = l}. In other words, we can identify an object in this fiber with a commutative diagram

Δ[0] z F

n p (8) D Δ[n] sd(N ).

On the other hand, an object in el(NF) over  can be identified with an n-chain of F F F¨ arrows z0 → ⋯ → zn in , lying over . The comparison functor u∶ el(N ) → sends such an n-chain to its last element, zn. Thus it suffices to show that there is a unique lift in Eq. (8). But this is exactly the condition that p is a right fibration.

Corollary 4.3. With notation as in Eqs. (6) and (7), we have natural isomorphisms

∗ ∗ ∗ )sd(ND)◦Qel(ND) ≅ sdND ◦)ND and Ntw(D) ≅ )sd(ND)◦Qel(ND)◦D.

Proof. Using that ) and el are inverse equivalences of categories between presheaves and right fibrations, these statements follow from Eqs. (6) and (7), right parts. D Lemma 4.4. For any decomposition space , there exists a functor untwD ∶ RFib∕tw(D) → Decomp∕D with untwD ◦ twD ≅ id . Decomp∕D Furthermore, both the left-to-right and the right-to-left squares commute up to natural isomor- phism: twD

Decomp∕D RFib∕tw(D) untwD ∗ ND D (9) elND

sSet∕ND RFib∕ el(ND). )ND

Proof. We already know that the left-to-right square commutes by Lemma 4.2 (left part). To define untwD making the right-to-left square commute, it suffices to show that for F D ∗ any right fibration p∶ → tw( ), the simplicial map )NDD(p) is CULF, because then it lands in Decomp (see the preliminaries on decomposition spaces, p.3). CULFness

11 amounts to certain squares being pullbacks. To express this cleanly, consider the right fibration q ∶ U → V defined by the following pullbacks:

F U ⌟ ⋅ ⌟ q p V el(ND) tw(D) ⌟ D

∆ ∆ active incl

The statement is that the right fibration q ∶ U → V corresponds to a cartesian natural transformation )U op ∆ ⇓ , active Set )V and this condition in turn can be read off directly on q: we need to check that for every active map f ∶ [m] ⇥ [n] the following square is a pullback of sets:

f ∗ Um Un

qm qn V V . m f ∗ n

 D We compare the qn-fiber over a point Δ[n] → in Vn with the qm-fiber over the Δ[f]  corresponding point Δ[m] → Δ[n] → D in Vm. We do this by computing these q-fibers in terms of p-fibers, using the explicit description of D from Remark 3.10. But we already noted in Remark 3.10 that D sends any arrow in el(ND) lying over an active D map to an identity, so the qn and qm fibers coincide for every  ∶ Δ[n] → and every active map f ∶ [m] ⇥ [n], and therefore the square is a pullback. Finally the main statement follows easily: we first read off from the commutativity of Eq. (9) that

∗ ND◦ untwD ◦ twD ≅ )ND◦D◦ twD ≅ )ND◦ elND ◦ND ≅ ND.

Since the nerve functor is fully faithful, so is the slice ND, and we have established untwD ◦ twD ≅ id Decomp∕D . Theorem 4.5 (Main theorem). For any decomposition space D, we have natural inverse equivalences of categories

twD Decomp∕D RFib∕tw(D). untwD

Proof. untwD ◦ twD ≅ id We established Decomp∕D in Lemma 4.4. For the other direction, since nerve is fully faithful, it suffices to prove that Ntw(D)◦ twD ◦ untwD ≅ Ntw(D); this

12 is the outer square in the diagram below:

Ntw(D) RFib∕tw(D) sSet∕ sd(ND) )sd(ND) ∗ 4.3 D RFib D RFib D ∕ el(N ) Q∗ ∕ el(sd N ) el(ND) sdND untwD 4.3 Ntw(D) 4.4 )ND sSet∕ND ND 4.1

Decomp D RFib D ∕ twD ∕tw( )

The up-to-iso commutativity of each of the squares inside was proven earlier as indi- cated.

Remark 4.6. All the proof ingredients are readily seen to be natural in D. In fact the main theorem can be seen as the D-component of a natural equivalence of Cat-valued functors Decomp∕− Decomp≃ Cat.

RFib∕tw(−)

Corollary 4.7. For D a decomposition space and F → tw(D) a right fibration, F is again the twisted arrow category of a decomposition space.

Corollary 4.8. The category Decomp is locally cartesian closed.

Bibliography

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J. Kock, Departament de Matemàtiques, Universitat Autònoma de Barcelona E-mail address: [email protected]

D.I. Spivak, Department of Mathematics, Massachusetts Institute of Technology E-mail address: [email protected]

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