Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the "homotopy theory" of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a "homotopy theory of homotopy theories." In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with their respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk's complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.
We prove that the homotopy theory of cofibration categories is equivalent to the homotopy theory of cocomplete quasicategories. This is achieved by presenting both homotopy theories as fibration categories and constructing an explicit equivalence between them.
These notes are from a series of lectures given at the Workshop on the Homotopy Theory of Homotopy Theories which took place in Caesarea, Israel, in May 2010. The workshop was organized by David Blanc, Emmanuel Farjoun, and David Kazhdan, and talks not indicated otherwise were given by the author.
Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks.
We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.
Generalizing a definition of homotopy fiber products of model categories, we give a definition of the homotopy limit of a diagram of left Quillen functors between model categories. As has been previously shown for homotopy fiber products, we prove that such a homotopy limit does in fact correspond to the usual homotopy limit, when we work in a more general model for homotopy theories in which they can be regarded as objects of a model category.
Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing `relations' into these universal gadgets. The paper develops this formalism and discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes.
We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of "stability relative to a class of functors", which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models has a well-behaved internal hom-object.
We introduce, for every positive integer n, the notion of an n-relative category and show that the category of the small n-relative categories is a model for the homotopy theory of n-fold homotopy theories, i.e. homotopy theories of ... of homotopy theories.
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $Γ$-categorie…
In this paper we extend Toën's derived Hall algebra construction, in which he obtains unital associative algebras from certain stable model categories, to one in which such algebras are obtained from more general stable homotopy theories, in particular stable complete Segal spaces satisfying appropriate finiteness assumptions.
We show that each of the three $K$-theory multifunctors from small permutative categories to $\mathcal{G}_*$-categories, $\mathcal{G}_*$-simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these $K$-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe that the right-induced homotopy theory of Bohmann-Osorno $\mathcal{E}_*$-categories is equivalent to the homotopy theory of pointed simplicial categories.
In previous work, the first author defined homotopy theories for stratified spaces from a simplicial and a topological perspective. In both frameworks stratified weak-equivalences are detected by suitable generalizations of homotopy links. These two frameworks are connected through a stratified version of the classical adjunction between the realization and the functor of singular simplices. Using a modified version of this adjunction, the first author showed that over a fixed poset of strata the two homotopy theories were equivalent. Building on this result we now show that the unmodified adj…
We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length. We study one extended example in detail: the theory of commutative monoids (which turns out to be essentially just a 2-category). This gives a straightforward, combinatorially explicit, and instructive notion of a commutative monoid. We prove that this definition is equivalent (in appropriate senses) both to the classical concept of an E-infinity monoid and to Lurie's concept of a commutative algebra object.
In this paper we establish new characterizations of stable derivators, thereby obtaining additional interpretations of the passage from (pointed) topological spaces to spectra and, more generally, of the stabilization. We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, and there are variants for sufficiently finite Kan extensions. As an additional reformulation, a derivator is stable if and only if it admits a zero object and if partial cone and partial fiber morphisms commute on squares.
We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n,X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord}(n,X), including the fact that it only depends on the (n+1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n+1 implies the existence of geometrical structures in X; in parti…
In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or "invertible" cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard mod…
Building on a previous definition of homotopy limit of model categories, we give a definition of homotopy colimit of model categories. Using the complete Segal space model for homotopy theories, we verify that this definition corresponds to the model-category-theoretic definition in that setting.
In this short note we provide a review of some developments in the area of homotopy quantum field theories, loosely based on a talk given by the second author at the Xth Oporto Meeting on Geometry, Topology and Physics.
For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev's 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories.
On développe une théorie de l'homotopie des 2-catégories analogue à la théorie de l'homotopie des catégories développée par Grothendieck dans "À la poursuite des champs". Il s'agit de la thèse de doctorat de l'auteur. We develop a homotopy theory of 2-categories analogous to Grothendieck's homotopy theory of categories developed in "Pursuing Stacks." This is the author's PhD thesis.
This survey offers an overview of an on-going project on uniform symmetries in abstract stable homotopy theories. This project has calculational, foundational, and representation-theoretic aspects, and key features of this emerging field on abstract representation theory include the following. First, generalizing the classical focus on representations over fields, it is concerned with the study of representations over rings, differential-graded algebras, ring spectra, and in more general abstract stable homotopy theories. Second, restricting attention to specific shapes, it offers an explanati…
Homotopy theories via the magnitude-path spectral sequence
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；CNRS, IMJ-PRG（法国国家科学研究中心，IMJ-PRG）；University of Edinburgh（爱丁堡大学）；University of Sheffield（谢菲尔德大学）
This series explores a new notion of T-homotopy equivalence of flows. The new definition involves embeddings of finite bounded posets preserving the bottom and the top elements and the associated cofibrations of flows. In this third part, it is proved that the generalized T-homotopy equivalences preserve the branching and merging homology theories of a flow. These homology theories are of interest in computer science since they detect the non-deterministic branching and merging areas of execution paths in the time flow of a higher dimensional automaton. The proof is based on Reedy model catego…
In the present paper we study bundles equipped with extra homotopy conditions, in particular so-called simplicial $n$-bundles. It is shown that (under some condition) the classifying space of 1-bundles is the double coset space of some finite dimensional Lie group. We also establish some relation between our bundles and C*-algebras.
We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and $Δ$-complexes, respectively. The functors of classifying spaces and face posets are compatible with these homotopy theories. In contrast with the classical settings of finite spaces and simplicial complexes, the universality of morphisms and simplices plays a central role in this paper.
In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong homotopy." Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of …
Over a monoidal model category, under some mild assumptions, we equip the categories of colored PROPs and their algebras with projective model category structures. A Boardman-Vogt style homotopy invariance result about algebras over cofibrant colored PROPs is proved. As an example, we define homotopy topological conformal field theories and observe that such structures are homotopy invariant.
In proper homotopy theory, the original concept of point used in the classical homotopy theory of topological spaces is generalized in order to obtain homotopy groups that study the infinite of the spaces. This idea: "Using any arbitrary object as base point" and even "any morphism as zero morphism" can be developed in most of the algebraic homotopy theories. In particular, categories with a natural cone have a generalized homotopy theory obtained through the relative homotopy relation. Generalized homotopy groups and exact sequences of them are built so that respective classical pointed ones …
Given an algebraic theory $\ct$, a homotopy $\ct$-algebra is a simplicial set where all equations from $\ct$ hold up to homotopy. All homotopy $\ct$-algebras form a homotopy variety. We give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi.
We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $Θ_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, the first author's $n$-fold complete Segal spaces, Kan and the first author's $n$-relative categories, and complete Segal space objects in any model of $(\infty,n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb{Z}/2)^n$.
We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative categories, not only with respect to their Reedy structures but also their Rezk structure, which turns them into models for the theories of (infinity, 1)- and (infinity, k)-categories, respectively.
We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.
An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict $T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed that for a suitable choice of T giving such an algebra amounts to providing the space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie algebra... Given a functor $X: T\to Spaces$ which preserves products up to a weak equivalence we show that $X$ is more or less canonically weakly equivalent to a strict $T$-algebra $LX$. Thus any `homotopy' algebrai…
In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinements of Khovanov homology: our refinement into a module over the connective k-theory spectrum and a stronger result by Lipshitz and Sarkar refining Khovanov homology into a stable homotopy type.
We introduce a linearly ordered lattice $μ(Grp)$ of torsion theories in simplicial groups. The torsion theories are defined where the torsion/torsion-free subcategories are given by the simplicial groups with bounded above/below Moore complex, respectively. These torsion theories extend naturally the torsion theories in internal groupoids in groups. Connections of this lattice with the homotopy groups are established since the homotopy groups of a simplicial group can be calculated as the quotients of torsion subojects.
Given a finite $\mathbb{Z}_2$-graded group $\hat{\mathsf{G}}$ with ungraded subgroup $\mathsf{G}$ and a twisted cocycle $\hatλ \in Z^n(B \hat{\mathsf{G}}; \mathsf{U}(1)_π)$ which restricts to $λ\in Z^n(B \mathsf{G}; \mathsf{U}(1))$, we construct a lift of $λ$-twisted $\mathsf{G}$-Dijkgraaf--Witten theory to an unoriented topological quantum field theory. Our construction uses a new class of homotopy field theories, which we call orientation twisted. We also introduce an orientation twisted variant of the orbifold procedure, which produces an unoriented topological field theory from an orientat…
We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.
Motivated by gauge theory, we develop a general framework for chain complex valued algebraic quantum field theories. Building upon our recent operadic approach to this subject, we show that the category of such theories carries a canonical model structure and explain the important conceptual and also practical consequences of this result. As a concrete application we provide a derived version of Fredenhagen's universal algebra construction, which is relevant e.g. for the BRST/BV formalism. We further develop a homotopy theoretical generalization of algebraic quantum field theory with a particu…
We check that there exists a model structure on the category of flows whose weak equivalences are the S-homotopy equivalences. As an application, we prove that the generalized T-homotopy equivalences preserve the branching and merging homology theories of a flow. The method of proof is completely different from the one of the third part of this series of papers.
This paper is part of an endeavor to define an analogue of the slice filtration in the unstable motivic homotopy category. Our approach was inspired by the fact that the triangulated structures do not play a relevant role for the construction of birational homotopy categories as well as by the work of Kahn-Sujatha \cite{K-theory/0596} on birational motives, where the existence of a connection between the layers of the slice filtration and birational invariants is explicitly suggested. Our main result, shows that there is an equivalence of categories between the orthogonal components for the sl…
Let k be an algebraically closed field of characteristic zero. Let SH(k) denote the motivic stable homotopy category of T-spectra over k and SH the classical stable homotopy category. Let c:SH -> SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism π_n(E)-> π_{n,0}c(E) for all spectra E. Fix an embedding of k into the complex numbers and let Re:SH(k) -> SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum has Betti realization which is…
The aim of this paper is to show that the most elementary homotopy theory of $\mathbf{G}$-spaces is equivalent to a homotopy theory of simplicial sets over $\mathbf{BG}$, where $\mathbf{G}$ is a fixed group. Both homotopy theories are presented as Relative categories. We establish the equivalence by constructing a strict homotopy equivalence between the two relative categories. No Model category structure is assumed on either Relative Category.
This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspo…
We quiver-interpret the classical simplicial theory - including the cosimplex category $Δ$, Dold-Kan correspondence, and Hochschild homology - as a certain Q-homotopy theory of type $A$. For the cyclic and cubical theories, we proceed analogously. Subsequently, we present far-reaching generalizations, using different types of quivers. Moreover, we explain how to construct certain categories as analogs of $Δ$, and associate to each a Q-homotopy theory. We provide many examples, including such theories of type $D$.
