Goncalo Tabuada
We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories.
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We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibration categories. Homotopy colimits for Quillen model categories are obtained as a particular case. We attach to each ABC cofibration category a left Heller derivator. A dual theory is developed for homotopy limits in ABC fibration categories and for right Heller derivators. These constructions provide a natural framework for 'doing homotopy theory' in ABC (co)fibration categories.
We construct a new Quillen model, based on the notions of Drinfeld's DG quotient and localization pair, for the Morita homotopy theory of DG categories. This new Quillen model carries a natural closed symmetric monoidal structure and allows us to re-interpret Toen's construction of the internal Hom-functor for the homotopy category of DG categories as a total right derived internal Hom-functor.
We give a new description of Rosenthal's generalized homotopy fixed point spaces as homotopy limits over the orbit category. This is achieved using a simple categorical model for classifying spaces with respect to families of subgroups.
In the last year of his life, Bob Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure which is inherited by functor categories. In this paper we explain and prove Thomason's results, based on his private notebooks. The first half presents Thomason's ideas about homotopy ends and its generalizations. This material may be of independent interest. Then we define Thomason model categories and give some examples. The usual proof shows that the homo…
For every regular cardinal $α$, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially DG categories which are stable under suspensions, cosuspensions, cones and $α$-small sums. Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.
We construct model category structures on various types of (marked) *-categories. These structures are used to present the infinity categories of (marked) *-categories obtained by inverting (marked) unitary equivalences. We use this presentation to explicitly calculate the \infty-categorical G-fixed points and G-orbits for G-equivariant (marked) *-categories.
We show that the free construction from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor. Our main result then shows that the induced functor between categories of algebras is an equivalence of homotopy theories. We describe an application to ring categories.
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 his paper [Ric89], Rickard presents the stable module category of a self-injective algebra as a Verdier quotient of its derived category by perfect complexes. We present a similar realization of the homotopy category in hopfological algebra as such a Verdier quotient.
In this paper we prove that Toen's derived enrichment of the model category of dg-categories defined by Tabuada, is computed by the dg-category of A-infinity functors. This approach was suggested by Kontsevich. We further put this construction into the framework of (infinity,2)-categories. Namely, we enhance the categories of dg and A-infinity categories, to (infinity,2)-categories. We prove that the (infinity,1)-truncation of to the (infinity,2)-category of dg-categories is a model for the simplicial localization at the model structure of Tabuada. As an application, we prove that the homotopy…
We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type $A_{2n}$ quivers.
We study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Marki and Tholen's semi-abelian categories. This model structure exists as soon as the base category A is regular Mal'tsev and has enough regular projectives; then the fibrations are the Kan fibrations of simplicial objects in A. When, moreover, A is semi-abelian, weak equivalences and homology isomorphisms coincide.
We prove that algebraic G-theory in is representable in unstable and stable motivic homotopy categories; in the stable category we identify it with the Borel-Moore theory associated to algebraic K-theory, and show that such an identification is compatible with the functorialities defined by Quillen and Thomason.
For every $\infty$-category $\mathscr{C}$, there is a homotopy $n$-category $\mathrm{h}_n \mathscr{C}$ and a canonical functor $γ_n \colon \mathscr{C} \to \mathrm{h}_n \mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy $n$-categories, we introduce the notion of an $n$-derivator and study the main examples arising from $\infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $\infty$-derivators and prov…
Let A be a dg category, F:A->A a dg functor inducing an equivalence of categories in degree-zero cohomology, and A/F the associated dg orbit category. For every A1-homotopy invariant (e.g. homotopy K-theory, K-theory with coefficients, etale K-theory and periodic cyclic homology), we construct a distinguished triangle expressing E(A/F) as the cone of the endomorphism E(F)-Id of E(A). In the particular case where F is the identity dg functor, this triangle splits and gives rise to the fundamental theorem. As a first application, we compute the A1-homotopy invariants of cluster (dg) categories, …
The main theorem in this paper is that the base change functor from a noetherian abelian category to its noetherian polynomial category induces an isomorphism on K-theory. The main theorem implies the well-known fact that A^1-homotopy invariance of K'-theory for noetherian schemes.
Algebraic $K_0$ for unpointed homotopy Categories
本文为无基点 Waldhausen 范畴和稳定 $\infty$-范畴引入 Grothendieck 堆概念，成功将代数 $K_0$ 研究扩展至无基点拓扑空间的同伦范畴。
Let $k$ be an algebraically closed field of exponential characteristic $p$. Given any prime $\ell\neq p$, we construct a stable étale realization functor $$\underline{\text{Ét}}_{\ell}:\text{Spt}(k)\rightarrow \text{Pro}(\text{Spt})^{H\mathbb{Z}/\ell}$$ from the stable $\infty$-category of motivic $\mathbb{P}^1$-spectra over $k$ to the stable $\infty$-category of $(H\mathbb{Z}/\ell)^*$-local pro-spectra (see section 3 for definition). This is induced by the étale topological realization functor á la Friedlander. The constant presheaf functor naturally induces the functor \[\text{SH}[1/p]\right…
We consider the effect of $t$-structures on the Tannaka duality theory for dg categories developed in our previous paper. We associate non-negative dg coalgebras $C$ to dg functors on the hearts of $t$-structures, and relate dg $C$-comodules to the original dg category. We give several applications for pro-algebraic homotopy types associated to various cohomology theories, and for motivic Galois groups.
A Computational Approach to the Homotopy Theory of DG categories
本文通过构造可计算的柱函子与同伦余极限函子，建立了半自由dg范畴及有限型dg范畴的余纤维化范畴，并应用于辛拓扑。
The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the d…
In this article we establish the foundations of the Morita homotopy theory of C*-categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure M_Mor on the category C*cat1 of small unital C*-categories. The weak equivalences are the Morita equivalences and the cofibrations are the *-functors which are injective on objects. As an application, we obtain an elegant description of the Brown-Green-Rieffel Picard group in the associated Morita homotopy category Ho(M_Mor). We then prove that the Morita homotopy category is semi-additive. By group c…
We prove that one can realize certain triangulated subcategories of the singularity category of a complete intersection as homotopy categories of matrix factorizations. Moreover, we prove that for any commutative ring and non-zerodivisor, the homotopy category of matrix factorizations embeds into the homotopy category of totally acyclic complexes of finitely generated projective modules over the factor ring.
This article provides some basic results on weight structures, weight complex functors and homotopy categories. We prove that the full subcategories K(A)^{w < n}, K(A)^{w > n}, K(A)^- and K(A)^+ (of objects isomorphic to suitably bounded complexes) of the homotopy category K(A) of an additive category A are idempotent complete, which confirms that (K(A)^{w <= 0}, K(A)^{w >= 0}) is a weight structure on K(A). We discuss weight complex functors and provide full details of an argument sketched by M. Bondarko, which shows that if w is a bounded weight structure on a triangulated category T that ha…
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