arXiv subjects

M. Szyld

Publications and source records attributed to M. Szyld.

5 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.

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arXiv2021

The Three F's for Bicategories I: Localization by Fractions is Exact

We study the interaction between the notions of filteredness, fractions and fibrations in the theory of bicategories, generalizing classical results for categories. We give an explicit formula for filtered pseudo-colimits of categories indexed by a bicategory, and we use it to compute the hom-categories of a bicategory of fractions. As a consequence, we show that the canonical pseudo-functor into a bicategory of fractions is exact.

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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.

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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.

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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.

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