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Register a new userThe category of Waldhausen categories as a closed multicategory
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Authors
Inna Zakharevich
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This paper works out in detail the closed multicategory structure of the category of Waldhausen categories.
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In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital dg categories over a field $Bbbk$. Our model structure can be thought of as an extension of the model structure on the category $mathrm{Cat}_{mathrm{dg}}(Bbbk)$ of (strictly unital) small dg categories over $Bbbk$, due to Tabuada [Tab]. More precisely, we show that the imbedding of $mathrm{Cat}_{mathrm{dg}}(Bbbk)$ to $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ is a right adjoint of a Quillen pair of functors. We prove that this Quillen pair is, in turn, a Quillen equivalence. In course of the proof, we study a non-symmetric dg operad $mathcal{O}$, governing the weakly unital dg categories, which is encoded in the Kontsevich-Soibelman definition. We prove that this dg operad is quasi-isomorphic to the operad $mathrm{Assoc}_+$ of unital associative algebras.
We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausens $S_{bullet}$-construction and therefore it defines a model for Waldhausen $K$-theory. As an example, we discuss this model for $A$-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausens $A(*)$. We also review the canonical map from the cobordism category of manifolds to $A$-theory from this viewpoint.
In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital dg categories (in the sense of Kontsevich-Soibelman [KS]) over a field $Bbbk$. In [PS], we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, saying that $mathrm{id}_xcdot mathrm{id}_x=mathrm{id}_x$ for any object $x$. Although this condition led us to a great simplification, it was redundant and had to be dropped. Here we get rid of this condition, and provide a closed model structure in full generality. The new closed model category is as well cofibrantly generated, and it is proven to be Quillen equivalent to the closed model category $mathrm{Cat}_mathrm{dg}(Bbbk)$ of (strictly unital) dg categories over $Bbbk$, given by Tabuada [Tab1]. Dropping the condition $mathrm{id}_x^2=mathrm{id}_x$ makes the construction of the closed model structure more distant from loc.cit., and requires new constructions. One of them is a pre-triangulated hull of a wu dg category, which in turn is shown to be a wu dg category as well. One example of a weakly unital dg category which naturally appears is the bar-cobar resolution of a dg category. We supply this paper with a refinement of the classical bar-cobar resolution of a unital dg category which is strictly unital (appendix B). A similar construction can be applied to constructing a cofibrant resolution in $mathrm{Cat}_mathrm{dgwu}(Bbbk)$.
In this article we build a Quillen model category structure on the category of sequentially complete l.m.c.-C*-algebras such that the corresponding homotopy classes of maps Ho(A,B) for separable C*-algebras A and B coincide with the Kasparov groups KK(A,B). This answers an open question posed by Mark Hovey about the possibility of describing KK-theory for C*-algebras using the language of Quillen model categories.
We prove that the Waldhausen Nil-group associated to a virtually cyclic groups that surjects onto the infinite dihedral group vanishes if and only if the corresponding Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses controlled pseudo-isotopy techniques of Farrell-Jones to establish the reverse implication.
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