Decomposition-space slices are toposes Joachim Kock ∗ David I. Spivak † Abstract We show that the category of decomposition spaces and CULF maps is locally a topos. Precisely, the slice category over any decomposition space D is a presheaf D topos, namely Decomp∕D ≃ Psh(tw ). 1 Introduction Decomposition spaces were introduced for purposesin combinatorics by Gálvez, Kock, and Tonks [7, 8, 9], and purposes in homological algebra, representation theory, and geometry by Dyckerhoff and Kapranov [4], who call them unital 2-Segal spaces. They are simplicial sets (or simplicial ∞-groupoids) with aproperty that expresses the ability to (co-associatively) decompose, just as in categories one can (associatively) compose. In particular, decomposition spaces induce coalgebras. The most nicely-behaved class of morphisms between decomposition spaces is that of CULF maps. These preserve decompositions in an appropriate way so as to induce coalgebra homomorphisms. Apart from the coalgebraic aspect, not so much is known about the category Decomp of decomposition spaces and CULF maps, and it may appear a bit peculiar. For exam- ple, the product of two decomposition spaces as simplicial sets is not the categorical product in Decomp—the projections generally fail to be CULF. Similarly, the termi- nal simplicial set (which is a decomposition space) is not terminal in Decomp. The simplicial-set product should rather be considered as a tensor product for decomposi- tion spaces. The present contribution advances the categorical study of decomposition spaces arXiv:1807.06000v3 [math.CT] 2 Sep 2019 by establishing that Decomp is locally a topos, meaning that all its slices are toposes— even presheaf toposes. More precisely we show: Main Theorem (Theorem 4.5). For D a decomposition space, there is a natural equivalence of categories ∼ D Decomp∕D → Psh(tw( )). (1) Here tw(D) is the twistedarrow category of D, obtained by edgewise subdivision—this is readily seen to be (the nerve of) a category, cf. Lemma 2.1 below. ∗ Kock was supported by grants MTM2016-80439-P (AEI/FEDER, UE) of Spain and 2017-SGR-1725 of Catalonia. † Spivak was supported by AFOSR grants FA9550–14–1–0031 and FA9550–17–1–0058. 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 <i<n); the inert maps are generated by the outer coface maps d⊥ = d0 and d⊤ = dn. (This orthogonal factorization system is an instance of the important general notion of generic-free factorization system of Weber [20] who referred to the two classes as generic and free. The active-inert terminology is due to Lurie [14].) 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¨.