A construction of certain weak colimits and an exactness property of the 2-category of categories
Theory and applications of categories, Tome 33 (2018), pp. 192-215.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

Given a 2-category $A$, a 2-functor $F : A \to Cat$ and a distinguished 1-subcategory $\Sigma \subset A$ containing all the objects, a $\sigma$-cone for $F$ (with respect to $\Sigma$) is a lax cone such that the structural 2-cells corresponding to the arrows of $\Sigma$ are invertible. The conical $\sigma$-limit} is the universal (up to isomorphism) $\sigma$-cone. The notion of $\sigma$-limit generalizes the well known notions of pseudo and lax limit. We consider the fundamental notion of $\sigma$-filtered pair $(A, \Sigma)$ which generalizes the notion of 2-filtered 2-category. We give an explicit construction of $\sigma$-filtered $\sigma$-colimits of categories, a construction which allows computations with these colimits. We then state and prove a basic exactness property of the 2-category of categories, namely, that $\sigma$-filtered $\sigma$-colimits commute with finite weighted pseudo (or bi) limits. An important corollary of this result is that a $\sigma$-filtered $\sigma$-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.
Publié le :
Classification : Primary: 18D05. Secondary: 18A30
Keywords: weak colimit, filtered, 2-category, exactness property
@article{TAC_2018_33_a7,
     author = {Descotte M.E. and Dubuc E.J. and Szyld M.},
     title = {A construction of certain weak colimits and an exactness property of the 2-category of categories},
     journal = {Theory and applications of categories},
     pages = {192--215},
     publisher = {mathdoc},
     volume = {33},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2018_33_a7/}
}
TY  - JOUR
AU  - Descotte M.E.
AU  - Dubuc E.J.
AU  - Szyld M.
TI  - A construction of certain weak colimits and an exactness property of the 2-category of categories
JO  - Theory and applications of categories
PY  - 2018
SP  - 192
EP  - 215
VL  - 33
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TAC_2018_33_a7/
LA  - en
ID  - TAC_2018_33_a7
ER  - 
%0 Journal Article
%A Descotte M.E.
%A Dubuc E.J.
%A Szyld M.
%T A construction of certain weak colimits and an exactness property of the 2-category of categories
%J Theory and applications of categories
%D 2018
%P 192-215
%V 33
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TAC_2018_33_a7/
%G en
%F TAC_2018_33_a7
Descotte M.E.; Dubuc E.J.; Szyld M. A construction of certain weak colimits and an exactness property of the 2-category of categories. Theory and applications of categories, Tome 33 (2018), pp. 192-215. http://geodesic.mathdoc.fr/item/TAC_2018_33_a7/