A Colimit Decomposition for Homotopy Algebras in Cat

Abstract

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 Rosický observed a key point to be that each homotopy colimit in SSet 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.

Rigidification of Algebras Over Essentially Algebraic Theories

“…On the other hand, by adding an object X 4 = X 4 1 to T , each homotopy model of the enlarged sketch is weakly equivalent to a model of H. Of course, this example leans on [6] as mentioned in 3.5.…”

“…Observe that the saturation does not change the flatness, i.e., Φ + = Φ * + . J. Bourke [6] proved that, over Cat, any cofibrant weight belongs to the iterative closure of finite coproducts of representables under colimits weighted by homotopy invariant Φ-flat weights. Thus there is a rigidification theorem for homotopy algebras in this case as well.…”