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Vertauschbarkeit von Limites und Colimites. (German) Zbl 0554.18001

A category \({\mathcal B}\) is called \({\mathcal A}\)-filtered for a given small category \({\mathcal A}\) when the functor \({\mathcal B}\to {\mathcal B}^{{\mathcal A}}\) is final. This generalizes the classical case of filtered categories by choosing \({\mathcal A}\) to be successively \(\emptyset\), \(\{\) 0,1\(\}\), \(\cdot \rightrightarrows \cdot\) or, which is equivalent, by choosing \({\mathcal A}\) to be any finite category.
It is well-known that in the category of Sets, finite limits commute with filtered colimits or, in other words, finite limits commute with \({\mathcal A}\)-filtered colimits where \({\mathcal A}\) is finite. The author studies the commutation in Sets of \({\mathcal A}\)-limits with \({\mathcal B}\)-colimits: a necessary condition is the \({\mathcal A}\)-filteredness of \({\mathcal B}\), but this condition is not sufficient.
The study developed here proves several commutation properties for special choices of \({\mathcal A}\) and \({\mathcal B}\). The general problem is also treated by splitting it in two cases: the consideration of discrete and connected \({\mathcal A}\); but the formal characterizations obtained (in terms of generalized completions) are probably not easy to handle in the practice.
Reviewer: F.Borceux

MSC:

18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
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References:

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