For a given category C and a topological space X, the constant stack on X with stalk C is the stack of locally constant sheaves with values in C. Its global objects are classified by their monodromy, a functor from the Poincare groupoid of X to C. In this paper we recall these notions from the point of view of higher category theory and then define the 2-monodromy of a locally constant stack with values in a 2-category as a 2-functor from the homotopy 2-groupoid into the 2-category. We show that 2-monodromy classifies locally constant stacks on a reasonably well-behaved space X. As an application, we show how to recover from this classification the cohomological version of a classical theorem of Hopf, and we extend it to the non abelian case.
Classical homological algebra considers chain complexes, resolutions, and derived functors in additive categories. We describe "track algebras in dimension n", which generalize additive categories, and we define higher order chain complexes, resolutions, and derived functors. We show that higher order resolutions exist in higher track categories, and that they determine higher order Ext-groups. In particular, the E_m-term of the Adams spectral sequence (m<n+3) is a higher order Ext-group, which is determined by the track algebra of higher cohomology operations.
In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully construct a category of abelian categories and suitably homotopically defined functors, and show that this category, equipped with the directional derivative, is a cartesian differential category in the sense of Blute, Cockett, and Seely. This provides an abstract framework that makes certain analogies between classical and functor calculus explicit. Inspired by Huang, Marcantognini, and Young's chain rule for higher order directional d…
One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when $n = 1$. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al. This pap…
In this article, it is shown that derivator six-functor-formalisms on any (classical) site canonically extend to higher geometric stacks as defined by Toën-Vezzosi under some natural locality conditions. As an application, it is shown that the six-functor-formalisms of Morel-Voevodsky-Ayoub extend to higher (Nisnevich-)Artin stacks locally of finite type over some fixed base scheme.
For a small $n$-category $\mathscr{C}$ and an $n$-topos $\mathscr{X}$, we study necessary and sufficient conditions for a functor $f \colon \mathscr{C} \to \mathscr{X}$ to determine a geometric morphism from $\mathscr{X}$ to the $n$-topos $\mathcal{P}(\mathscr{C})_n$ of presheaves on $\mathscr{C}$ for any $n \geq 1$. These results generalize and unify results of Lurie for $n=\infty$ and classical characterizations of flat functors (Diaconescu's theorem) for $n=1$. Interestingly, for $n=\infty$, our analogue of Diaconescu's theorem requires hypercompleteness. As an application, we show that the…
We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\infty$-categories if $n \geq 2$, and the homotopy $n$-categories of presentable stable $\in…
This is a sequel to a previous paper, developing an intrinsic, combinatorial homotopy theory for simplicial complexes; the latter form the cartesian closed subcategory of 'simple presheaves' in !Smp, the topos of symmetric simplicial sets, or presheaves on the category of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself, !Smp. As a crucial advantage, the fundamental groupoid functor !Smp --> Gpd is left adjoint to a natural functor Gpd --> !Smp, the symmetric nerve of a groupoid, and preserves all colimits - a strong van Kampen property. Sim…
We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techni…
The main objective of this paper is to construct a homotopy colimit functor on a category of functors taking values in the model category of quasi-categories.
The branching (resp. merging) space functor of a flow is a left Quillen functor. The associated derived functor allows to define the branching (resp. merging) homology of a flow. It is then proved that this homology theory is a dihomotopy invariant and that higher dimensional branchings (resp. mergings) satisfy a long exact sequence.
We determine a family of functors from a poset to abelian groups such that the higher direct limits vanish on them. This is done by first characterizing the projective functors. Then a spectral sequence arising from the grading of the poset is used. Also the dual version for injective functors and higher inverse limits is included. Graded posets include simplicial complexes, subdivision categories and simplex-like posets.
The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled n-cube, in exactly one way. The main ingredient is the non-functorial construction called labelled directed coskeleton. It is defined as a subobject of the labelled coskeleton, the latter coinciding in the unlabelled case with the right adjoint to the truncation functor. This non-functorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirem…
We give a new criterion guaranteeing existence of model structures left-induced along a functor admitting both adjoints. This works under the hypothesis that the functor induces idempotent adjunctions at the homotopy category level. As an application, we construct new model structures on cubical sets, prederivators, marked simplicial sets and simplicial spaces modeling $\infty$-categories and $\infty$-groupoids.
We develop a theory of Goodwillie calculus for functors between $G$-equivariant homotopy theories, where $G$ is a finite group. We construct $J$-excisive approximations of a homotopy functor for any finite $G$-set $J$. These fit together into a poset, the Goodwillie tree, that extends the classical Goodwillie tower. We prove convergence results for the tree of a functor on pointed $G$-spaces that commutes with fixed-points, and we reinterpret the Tom Dieck-splitting as an instance of a more general splitting phenomenon that occurs for the fixed-points of the equivariant derivative of these fun…
The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $\mathcal{E}$ is a presentable $\infty$-category, then we give a concrete formula for the constant hypersheaf functor $\mathcal{E}\to…
This text is based on lectures given by authors in summer 2015. It contains an introduction to the theory of limits over the category of presentations, with examples of different well-known functors like homology or derived functors of non-additive functors in a form of derived limits. The theory of so-called ${\bf fr}$-codes also is developed. This is a method how different functors from the category of groups to the category of abelian groups, such as group homology, tensor products of abelianization, can be coded as sentences in the alphabet with two symbols ${\bf f}$ and ${\bf r}$.
We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct…
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor $\infty$-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straight…
We prove that each exponential functor on the category of finite-dimensional complex inner product spaces and isomorphisms gives rise to an equivariant higher (ie. non-classical) twist of $K$-theory over $G=SU(n)$. This twist is represented by a Fell bundle $\mathcal{E} \to \mathcal{G}$, which reduces to the basic gerbe for the top exterior power functor. The groupoid $\mathcal{G}$ comes equipped with a $G$-action and an augmentation map $\mathcal{G} \to G$, that is an equivariant equivalence. The $C^*$-algebra $C^*(\mathcal{E})$ associated to $\mathcal{E}$ is stably isomorphic to the section …
A lot of well-known functors such as group homology, cyclic homology of algebras can be described as limits of certain simply defined functors over categories of presentations. In this paper, we develop technique for the description of the higher limits over categories of presentations and show that certain homological functors can be described in this way. In particular, we give a description of Hochschild homology and the derived functors of tensor, symmetric and exterior powers in the sense of Dold and Puppe as higher limits.
Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In this paper we study the more complicated setup where we have two pushforwards (an "additive" and a "multiplicative" one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\infty$-cat…
We present a novel approach to the problem of integrating homotopy Lie algebras by representing the Maurer-Cartan space functor with a universal cosimplicial object. This recovers Getzler's original functor but allows us to prove the existence of additional, previously unknown, structures and properties. Namely, we introduce a well-behaved left adjoint functor, we establish functoriality with respect to infinity-morphisms, and we construct a coherent hierarchy of higher Baker-Campbell-Hausdorff formulas. Thanks to these tools, we are able to establish the most important results of higher Lie t…
In the spirit of Morse homology initiated by Witten and Floer, we construct two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one $\mathcal{A}$ comes out of the Morse-Samle pairs and their higher homotopies, and the strict one $\mathcal{B}$ concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak $\infty$-functor $\mathcal{F}: \mathcal{A}\rightarrow \mathcal{B}$. Higher algebraic structures behind Morse homology are revealed with the perspe…
We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous $\infty$-category. The heart of the proof is the description of the $\infty$-category of $\mathcal C$-valued sheaves on $\mathcal X$ as an $\infty$-category of functors that satisfy finite limits conditions as well as filtered colimits conditions (instead of limits conditions purely); we call such functors $ω$-continuous sheaves. As an application, we show that when $\mathcal X$ is exponentiable, its $\infty$-category of stable sheaves $\m…
We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on…
We study the higher derived functors of the inverse limit of a functor F: D --> Z_{(p)}-mod, where D is one of the standard categories which arise when studying the homotopy theory of the classifying space of a finite group G, e.g., the orbit category or the Quillen category of G. These higher limits are of importance e.g., for the study of maps between classifying spaces as well as for group cohomology. We show that these higher limits can be identified with the G-equivariant Bredon cohomology of the subgroup complex of p-subgroups in G (i.e., the nerve of the poset of p-subgroups in G) with …
We construct a family of oriented extended topological field theories using the AKSZ construction in derived algebraic geometry, which can be viewed as an algebraic and topological version of the classical AKSZ field theories that occur in physics. These have as their targets higher categories of symplectic derived stacks, with higher morphisms given by iterated Lagrangian correspondences. We define these, as well as analogous higher categories of oriented derived stacks and iterated oriented cospans, and prove that all objects are fully dualizable. Then we set up a functorial version of the A…
We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (inf…
