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Register a new userThe DG-category of secondary cohomology operations
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We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of the first author on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.
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When $mathbb C$ is a semi-abelian category, it is well known that the category $mathsf{Grpd}(mathbb C)$ of internal groupoids in $mathbb C$ is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property of being semi-abelian is replaced by the one of being action representable (in the sense of Borceux, Janelidze and Kelly) turns out to be rather subtle. In the present article we give a sufficient condition for this to be true: in fact we prove that the category $mathsf{Grpd}(mathbb C)$ is a semi-abelian action representable algebraically coherent category with normalizers if and only if $mathbb C$ is a semi-abelian action representable algebraically coherent category with normalizers. This result applies in particular to the categories of internal groupoids in the categories of groups, Lie algebras and cocommutative Hopf algebras, for instance.
In this short paper we outline (mostly without proofs) our new approach to the derived category of sheaves of commutative DG rings. The proofs will appear in a subsequent paper. Among other things, we explain how to form the derived intersection of two closed subschemes inside a given algebraic scheme X, without recourse to simplicial or higher homotopical methods, and without any global assumptions on X.
Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris-Rips complex in applied topology. We formalize a remedy by introducing a category of simplicial metric thickenings whose objects have a natural realization as metric spaces. The properties of this category allow us to prove that, for a large class of thickenings including Vietoris-Rips and Cech thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.
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 $mathrm{Cat}_mathrm{dgwu}(Bbbk)$.
We show that the category of positive opetopes with contraction morphisms, i.e. all face maps and some degeneracies, forms a test category. The category of positive opetopic sets pOpeSet can be defined as a full subcategory of the category of polygraphs Poly. An object of pOpeSet has generators whose codomains are again generators and whose domains are non-identity cells (i.e. non-empty composition of generators). The category pOpeSet is a presheaf category with the exponent being called the category of positive opetopes pOpe. Objects of pOpe are called positive opetopes and morphisms are face maps only. Since Poly has a full-on-isomorphisms embedding into the category of omega-categories oCat, we can think of morphisms in pOpe as omega-functors that send generators to generators. The category of positive opetopes with contractions pOpe_iota has the same objects and face maps pOpe, but in addition it has some degeneracy maps. A morphism in pOpe_iota is an omega-functor that sends generators to either generators or to identities on generators. We show that the category pOpe_iota is a test category.
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