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A lattice-theoretical perspective on adhesive categories. (English) Zbl 1217.18004

First roughly some definitions: A natural transformation \(\beta :F\Rightarrow G:{\mathcal C}\to{\mathcal D}\) between two functors \( F,G\) is cartesian if the usual commuting squares are all pullbacks.
Let \(D:{\mathcal I}\to{\mathcal C}\) be a diagram. A cocone for \(D\) is a natural transformation \(\varphi ^{A}:D\Rightarrow A\) to a constant functor \(A:{\mathcal I}\to{\mathcal C}\). A colimit for \(D\) is a cocone \(\varphi^A\) defined by the natural universality property. Such a colimit is a Van Kampen colimit if – given another diagram \(D':{\mathcal I}\to{\mathcal C}\), a cocone \(\varphi ^{B}:D'\to B\), a cartesian natural transformation \( \beta :D\Rightarrow D'\) and a morphism \(s:B\to A\) in \({\mathcal C}\) which commute – it holds that \(\varphi^B\) is a colimit if and only if the commuting square associated in \({\mathcal C}\) is a pullback.
Finally, a category \({\mathcal C}\) is called adhesive if it has pullbacks, pushouts along monos and pushouts along monos are Van Kampen pushouts.
The subobjects in an adhesive category form a distributive lattice. After defining the notion of irreducible object in adhesive categories it is proved that any (finite) object can be obtained as the colimit of its irreducible subobjects, and this is a Van Kampen colimit.
Applications of representation theory for distributive lattices are also given.

MSC:

18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
06D05 Structure and representation theory of distributive lattices
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References:

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