arXiv subjects

M. E. Descotte

Publications and source records attributed to M. E. Descotte.

4 recordsLinked to original sources

arXiv2022

Model bicategories and their homotopy bicategories

We give the definitions of model bicategory and $q$-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory $\mathcal{C}$, denote by $\mathcal{C}_{fc}$ the full sub-bicategory of the fibrant-cofibrant objects. We prove that the 2-dimensional localization of $\mathcal{C}$ at the weak equivalences can be computed as a bicategory $\mathcal{H}o(\mathcal{C})$ whose objects and arrows are those of $\mathcal{C}_{fc}$ and whose 2-cells are classes of $q$-homotopies up to an equivalence relation. When considered for a model category, $q$-homotopies coincide with the homotopies as considered by Quillen. The pseudofunctor $\mathcal{C} \stackrel{q}{\longrightarrow} \mathcal{H}o(\mathcal{C})$ which yields the localization is constructed by using a notion of fibrant-cofibrant replacement in this context. We include an appendix with a general result of independent interest on a transfer of structure for lax functors, that we apply to obtain a pseudofunctor structure for the fibrant-cofibrant replacement.

math.CT↗

arXiv2021

A localization of bicategories via homotopies

Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C,W) that allow us to construct the bicategorical localization with respect to W by dealing only with the 2-cells, that is without adding objects or arrows to C. We show that in this case, the 2-cells of the localization can be given by the homotopies with respect to W, a notion defined in this article which is closely related to Quillen's notion of homotopy for model categories but depends only on a single family of arrows. This localization result has a natural application to the construction of the homotopy bicategory of a model bicategory, which we develop elsewhere, as the pair (C_{fc},W) given by the weak equivalences between fibrant-cofibrant objects satisfies the conditions given in the present article.

math.CT↗

arXiv2018

Sigma limits in 2-categories and flat pseudofunctors

In this paper we introduce sigma limits (which we write $σ$-limits), a concept that interpolates between lax and pseudolimits: for a fixed family $Σ$ of arrows of a 2-category $\mathcal{A}$, a $σ$-cone for a $2$-functor $\mathcal{A} \stackrel{F}{\rightarrow} \mathcal{B}$ is a lax cone such that the structural 2-cells corresponding to the arrows of $Σ$ are invertible. The conical $σ$-limit of $F$ is the universal $σ$-cone. Similary we define $σ$-natural transformations and weighted $σ$-limits. We consider also the case of bilimits. We develop the theory of $σ$-limits and $σ$-bilimits, whose importance relies on the following key fact: any weighted $σ$-limit (or $σ$-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary $\mathcal{C}at$-valued 2-functor as a conical $σ$-bicolimit of representable 2-functors, for a suitable choice of $Σ$, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a $\mathcal{C}at$-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class $Σ$, which we call $σ$-filtered. Our main result is: A pseudofunctor $\mathcal{A} \rightarrow \mathcal{C}at$ is flat if and only if it is a $σ$-filtered $σ$-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.

math.CT↗

arXiv2016

A construction of certain weak colimits and an exactness property of the 2-category of categories

Given a 2-category $\mathcal{A}$, a $2$-functor $\mathcal{A} \overset {F} {\longrightarrow} \mathcal{C}at$ and a distinguished 1-subcategory $Σ\subset \mathcal{A}$ containing all the objects, a $σ$-cone for $F$ (with respect to $Σ$) is a lax cone such that the structural $2$-cells corresponding to the arrows of $Σ$ are invertible. The conical $σ$-limit is the universal (up to isomorphism) $σ$-cone. The notion of $σ$-limit generalises the well known notions of pseudo and lax limit. We consider the fundamental notion of $σ$-filtered} pair $(\mathcal{A}, \, Σ)$ which generalises the notion of 2-filtered 2-category. We give an explicit construction of $σ$-filtered $σ$-colimits of categories, construction which allows computations with these colimits. We then state and prove a basic exactness property of the 2-category of categories, namely, that $σ$-filtered $σ$-colimits commute with finite weighted pseudo (or bi) limits. An important corollary of this result is that a $σ$-filtered $σ$-colimit of exact category valued 2-functors is exact. This corollary is essential in the 2-dimensional theory of flat and pro-representable 2-functors, that we develop elsewhere.

math.CT↗