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

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Descotte M.E., Dubuc E.J., Szyld M.

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.

Keywords:
weak colimit, filtered, 2-category, exactness property

2010 MSC:
Primary: 18D05. Secondary: 18A30

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 8, pp 192-215.

Published 2018-03-12.

http://www.tac.mta.ca/tac/volumes/33/8/33-08.pdf

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