We construct a nerve from double categories into double $(\infty,1)$-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories and the model structure on bisimplicial spaces for double $(\infty,1)$-categories seen as double Segal objects in spaces complete in the horizontal direction. We then restrict the nerve along a homotopical horizontal embedding of 2-categories into double categories, and show that it gives a right Quillen and homotopically fully faithful functor between Lack's model structure for 2-categories and the model structure for 2-fold complete Segal spaces. We further show that Lack's model structure is right-induced along this nerve from the model structure for 2-fold complete Segal spaces.
For most models of $(\infty,2)$-categories an embedding of the $\infty$-category of 2-categories into that of $(\infty,2)$-categories has been constructed in the form of a nerve construction of some flavor. We prove that all those nerve embeddings induce equivalent functors, modulo change of model. We also show that all the nerve embeddings realize the $\infty$-category of 2-categories as the sub-$\infty$-category of $(\infty,2)$-categories that are local with respect to a certain class of maps.
Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the classes of $n$-categorical weak equivalences that they define coincide: they are the Thomason equivalences. We give two applications of this result: the first one states that a Dyer-Kan-type equivalence for Thomason equivalences is a Thomason equivalence; the second one, fundamental, is the stability of the class of Thomason equivalences under the dualities of the…
The most natural notion of a simplicial nerve for a (weak) bicategory was given by Duskin, who showed that a simplicial set is isomorphic to the nerve of a $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if and only if it is a quasicategory which has unique fillers for inner horns of dimension $3$ and greater. Using Duskin's technique, we show how his nerve applies to $(2,1)$-category functors, making it a fully faithful inclusion of $(2,1)$-categories into simplicial sets. Then we consider analogues of this extension of Duskin's result for several different two-dimensional …
We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's model structure for 2-categories and Riehl-Verity's model structure for 2-complicial sets. Furthermore, we show that Lack's model structure is transferred along this nerve and that the nerve is homotopically fully faithful.
In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to $\infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $\infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.
We prove a certain proposition which states a relationship between coverings of small categories and nerves. As its application, we prove that for a covering $\map{P}{E}{B}$ of finite categories, the zeta function of $E$ is the zeta function of $B$ to the number of sheet of $P$. Moreover, we prove the formula $χ(E)=χ(F)χ(B)$ for Euler characteristic of categories and coverings.
We give an explicit and purely combinatorial description of the Duskin nerve of any (r+1)-point suspension 2-category, and in particular of any 2-category belonging to Joyal's cell category Theta_2.
In this paper we prove that various quasi-categories whose objects are $\infty$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit…
This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of homotopy theory and the Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier and Porter's homotopy coherent nerve, and show that this faithfully represents complicial Gray-categories as weak complicial sets. The category of weak complicial sets may itself be canonically enriched to a compl…
In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a relative category that has the two out of six property and admits a 3-arrow calculus. We then show that Charles Rezk's result that the simplicial space obtained from a simplicial model category by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space also holds for these partial model categories. We also note that conversely ever…
We functorially associate to each relative $\infty$-category $(R,W)$ a simplicial space $N^R_\infty(R,W)$, called its Rezk nerve (a straightforward generalization of Rezk's "classification diagram" construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve $N^R_\infty(R,W)$ is precisely the one corresponding to the localization $R[[W^{-1}]]$; and (ii) that the Rezk nerve functor defines an equivalence $RelCat_\infty [[ W_{BK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a…
The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's category $Θ_2$. We prove that the nerve of a bicategory is a $2$-quasi-category (a model for $(\infty,2)$-categories due to Ara), and moreover that the nerve functor restricts to the right part of a Quillen equivalence between Lack's model structure for bicategories and a Bousfield localisation of Ara's model structure for $2$-quasi-categories. We deduce that Lack'…
In this paper we introduce a functor, called the simplicial nerve of an A-infinity category, defined on the category of (small) A-infinity categories with values in simplicial sets. We prove that the simplicial nerve of any A-infinity category is an infinity category. This construction extends functorially the nerve construction for differential graded categories proposed by J.Lurie in Higher Algebra. We prove that if a differential graded category is pretriangulated in the sense of A.I.Bondal-M.Kapranov, then its nerve is a stable infinity category in the sense of J.Lurie.
We give a direct proof that middle mapping spaces in coherent nerves of Kan enriched categories have the same homotopy type as the original mapping spaces.
Let $I$ be a small category with finite dimensional nerve, and $X\colon I\to Cat$ a diagram of small categories. We show that, under a "Reedy quasi-fibrancy condition", the homotopy limit of the geometric realization of $X$ is itself the geometric realization of a category. This categorical model for the homotopy limit is defined explicitly, as a category of natural transformations of diagrams. For the poset $\bullet\to\bullet\leftarrow\bullet$ we recover the model for homotopy pullbacks provided by Quillen's Theorem $B$ (specifically Barwick and Kan's version of Quillen's Theorem $B_2$). For …
We present a combinatorial analogue of the nerve theorem for covers of small categories, using the Grothendieck construction. We apply our result to prove the inclusion-exclusion principle for the Euler characteristic of a finite category.
The classifying diagram was defined by Rezk and is a generalization of the nerve of a category; in contrast to the nerve, the classifying diagram of two categories is equivalent if and only if the categories are equivalent. In this paper we prove that the classifying diagram of any category is characterized in terms of classifying spaces of stabilizers of groups. We also prove explicit decompositions of the classifying diagrams for the categories of finite ordered sets, finite dimensional vector spaces, and finite sets in terms of classifying spaces of groups.
This paper corrects a small mistake in a paper of Dwyer-Kan, and uses this to identify homotopy function complexes in a model category with the nerves of certain categories of zig-zags.
The paper focuses on investigating how certain relations between strict $n$-categories are preserved in a particular implementation of $(\infty,n)$-categories, given by saturated $n$-complicial sets. In this model, we show that the $(\infty,n)$-categorical nerve of $n$-categories is homotopically compatible with $1$-categorical suspension and wedge. As an application, we show that certain pushouts encoding composition in $n$-categories are homotopy pushouts of saturated $n$-complicial sets.
We prove the compatibility between the suspension construction and the complicial nerve of $ω$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete Segal $Θ_n$-spaces and Verity's $n$-complicial sets.
We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. This is an n-fold analogue to Thomason's Quillen model structure on Cat. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, we completely prove that the un…
In this paper we obtain several model structures on {\bf DblCat}, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in {\bf Cat} and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, {\bf DblCat} inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements …
We construct a generalization of the operadic nerve, providing a translation between the equivariant simplicially enriched operadic world to the parametrized $\infty$-categorical perspective. This naturally factors through genuine equivariant operads, a model for "equivariant operads with norms up to homotopy". We introduce the notion of an op-fibration of genuine equivariant operads, extending Grothendieck op-fibrations, and characterize fibrant operads as the image of genuine equivariant symmetric monoidal categories. Moreover, we show that under the operadic nerve, this image is sent to $G$…
The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint C are important to the study of (infinity,1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor C also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category CX arising from a quasi-category X are not well understood. We show that when X is…
We consider four categories: the category of diagrams of small categories indexed by a given small category O, the (comma) category of small categories over O, the category of diagrams of simplicial sets indexed by O, and the category of simplicial sets over the nerve of O. Fritsch and Golasinski claimed that these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak equivalences are the ``weak fibre homotopy e…
This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the current paper, we give a conceptual proof of an $\infty$-categorical nature of the same theorem. This proof is based on the theory of join and slices for strict $\infty$-categories developed by the authors in a previous paper, and on a comma construction for strict $\infty$-categories generalizing classical comma categories and Gray's comma 2-categories. This $…
Any tricategory characteristically has associated various simplicial or pseudo-simplicial objects. This paper explores the relationship amongst three of them: the pseudo-simplicial bicategory so-called Grothendieck nerve of the tricategory, the simplicial bicategory termed its Segal nerve, and the simplicial set called its Street geometric nerve, and it proves the fact that the geometric realizations of all of these possible candidate 'nerves of the tricategory' are homotopy equivalent. Our results provide coherence for all reasonable extensions to tricategories of Quillen's definition of the …
We put a model structure on the category of categories internal to simplicial sets whose weak equivalences are reflected by the nerve functor to bisimplicial sets with Rezk's model structure. This model structure is shown to be Quillen equivalent to Rezk's model structure, thus internal category are another model for $(\infty,1)$-categories.
We propose a categorification of the Dowker duality theorem for relations. Dowker's theorem states that the Dowker complex of a relation $R \subseteq X \times Y$ of sets $X$ and $Y$ is homotopy equivalent to the Dowker complex of the transpose relation $R^T \subseteq Y \times X$. Given a relation $R$ of small categories $\mathcal{C}$ and $\mathcal{D}$, that is, a functor of the form $R \colon \mathcal{R} \to \mathcal{C} \times \mathcal{D}$, we define the bisimplicial rectangle nerve $ER$ and the Dowker nerve $DR$. The diagonal $d(ER)$ of the bisimplicial set $ER$ maps to the simplicial set $DR…
This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate `nerves of C' are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major result proves a direct extension of Thomason's `Homotopy Colimit Theorem' to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying spa…
In this paper we describe two ways on which cofibred categories give rise to bisimplicial sets. The "fibred nerve" is a natural extension of Segal's classical nerve of a category, and it constitutes an alternative simplicial description of the homotopy type of the total category. If the fibration is splitting, then one can construct the "cleaved nerve", a smaller variant which emerges from a distinguished closed cleavage. We interpret some classical theorems by Thomason and Quillen in terms of our constructions, and use the fibred and cleaved nerve to establish new results on homotopy and homo…
The article has been withdrawn - the proof of the final corollary turned out to rely on an unproven statement. If the authors will manage to repair the argument, resubmission will be made.
We propose sparse versions of filtered simplicial complexes used to compte persistent homology of point clouds and of networks. In particular we extend a slight variation of the Sparse Čech Complex of Cavanna, Jahanseir and Sheehy from point clouds in Cartesian space to point clouds in arbitrary metric spaces. Along the way we formulate interleaving in terms of strict $2$-categories, and we introduce the concept of Dowker dissimilarities that can be considered as a common generalization of metric spaces and networks.
Infinity-categorical universal properties of quotients and localizations of A-infinity-categories
Center for Geometry and Physics (IBS), Pohang, Korea（韩国浦项几何与物理中心（IBS））；Department of Mathematics, POSTECH, Pohang Korea（韩国浦项科学技术大学数学系）；Department of Mathematics, Texas State University（德克萨斯州立大学数学系）
This paper answers the following question: what algebraic structure on a category corresponds to an $A_n$ structure (in the sense of Stasheff) on the geometric realization of its nerve?
Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types …
This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double cate…
We study the problem of when triangulated categories admit unique infinity-categorical enhancements. Our results use Lurie's theory of prestable infinity-categories to give conceptual proofs of, and in many cases strengthen, previous work on the subject by Lunts--Orlov and Canonaco--Stellari. We also give a wide range of examples involving quasi-coherent sheaves, categories of almost modules, and local cohomology to illustrate the theory of prestable infinity-categories. Finally, we propose a theory of stable $n$-categories which would interpolate between triangulated categories and stable inf…
The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. This paper describes how these facts are related to a categorical analogue of topological delooping. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a…
Let $ A $ be a finite connected graded cocommutative Hopf algebra over a field $ k $. There is an endofunctor $ \mathsf{tw} $ on the stable module category $ \mathrm{StMod}_A $ of $ A $ which twists the grading by $ 1 $. We show the categorical entropy of $ \mathsf{tw} $ is zero. We provide a lower bound for the categorical polynomial entropy of $ \mathsf{tw} $ in terms of the Krull dimension of the cohomology of $ A $, and an upper bound in terms of the existence of finite resolutions of $ A $-modules of a particular form. We employ these tools to compute the categorical polynomial entropy of…
