arXiv subjects

E. J. Dubuc

Publications and source records attributed to E. J. Dubuc.

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

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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arXiv2000

On the Galois Theory of Grothendieck

In this paper we deal with Grothendieck's interpretation of Artin's interpretation of Galois's Galois Theory (and its natural relation with the fundamental group and the theory of coverings) as he developed it in Expose V, section 4, ``Conditions axiomatiques d'une theorie de Galois'' in the SGA1 1960/61. This is a beautiful piece of mathematics very rich in categorical concepts, and goes much beyond the original Galois's scope (just as Galois went much further than the non resubility of the quintic equation). We show explicitly how Grothendieck's abstraction corresponds to Galois work. We introduce some axioms and prove a theorem of characterization of the category (topos) of actions of a discrete group. This theorem corresponds exactly to Galois fundamental result. The theorem of Grothendieck characterizes the category (topos) of continuous actions of a profinite topological group. We develop a proof of this result as a "passage into the limit'' (in an inverse limit of topoi) of our theorem of characterization of the topos of actions of a discrete group. We deal with the inverse limit of topoi just working with an ordinary filtered colimit (or union) of the small categories which are their (respective) sites of definition. We do not consider generalizations of Grothendieck's work, except by commenting briefly in the last section how to deal with the prodiscrete (not profinite) case. We also mention the work of Joyal-Tierney, which falls naturally in our discussion. There is no need of advanced knowledge of category theory to read this paper, exept for the comments in the last section.

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