A colimit decomposition for homotopy algebras in Cat

Badzioch showed that in the category of simplicial sets each homotopy algebra of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to extend this result to other contexts Rosicky observed a key point to be that each homotopy colimit in simplicial sets admits a decomposition into a homotopy sifted colimit of finite coproducts, and asked the author whether a similar decomposition holds in the 2-category of categories Cat. Our purpose in the present paper is to show that this is the case.Comment: Some notation changed; small amount of exposition added in intr

Noncommutativity as a colimit

Every partial algebra is the colimit of its total subalgebras. We prove this result for partial Boolean algebras (including orthomodular lattices) and the new notion of partial C*-algebras (including noncommutative C*-algebras), and variations such as partial complete Boolean algebras and partial AW*-algebras. The first two results are related by taking projections. As corollaries we find extensions of Stone duality and Gelfand duality. Finally, we investigate the extent to which the Bohrification construction, that works on partial C*-algebras, is functorial.Comment: 22 pages; updated theorem 15, added propoisition 3

On the equvialence of colimits and 2-colimits

We compare the colimit and 2-colimit of strict 2-functors in the 2-category of groupoids, over a certain type of posets. These posets are of special importance, as they correspond to coverings of a topological space. The main result of this paper gives conditions on the 2-functor $\mathfrak{F}$, for which $\mathsf{colim}\mathfrak{F}\simeq2\mathsf{colim}\mathfrak{F}$. One can easily see that any 2-functor $\mathfrak{F}$ can be deformed to a 2-functor $\mathfrak{F}'$, which satisfied the conditions of the theorem