For small categories over a fixed object set, cohomology can be computed via direction functors on S-special morphisms, matching Hoff and Golasinski cohomology in every degree.
Partial Mal'tsevness and partial protomodularity
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abstract
We introduce the notion of Mal'tsev reflection which allows us to set up a partial notion of Mal'tsevness with respect to a class $\Sigma$ of split epimorphisms stable under pullback and containing the isomorphisms, and we investigate what is remaining of the properties of the global Mal'tsev context. We introduce also the notion of partial protomodularity in the non-pointed context.
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A direction functor approach to the cohomology of small categories
For small categories over a fixed object set, cohomology can be computed via direction functors on S-special morphisms, matching Hoff and Golasinski cohomology in every degree.