Shrinkability, relative left properness, and derived base change

TL;DR

This paper establishes conditions under which categories of colored props for certain pasting schemes admit model structures with desirable properties and explores how these structures are preserved under base category equivalences.

Contribution

It introduces the concept of shrinkability in pasting schemes and shows how it ensures model structures are (weakly) left proper and are preserved under Quillen equivalences.

Findings

Model structures exist for categories of props under certain conditions.

Shrinkability of pasting schemes implies weak left properness.

Quillen equivalences between base categories induce equivalences between prop categories.

Abstract

For a connected pasting scheme $\mathcal G$, under reasonable assumptions on the underlying category, the category of $\mathfrak C$-colored $\mathcal G$-props admits a cofibrantly generated model category structure. In this paper, we show that, if $\mathcal G$ is closed under shrinking internal edges, then this model structure on $\mathcal G$-props satisfies a (weaker version) of left properness. Connected pasting schemes satisfying this property include those for all connected wheeled graphs (for wheeled properads), wheeled trees (for wheeled operads), simply connected graphs (for dioperads), unital trees (for symmetric operads), and unitial linear graphs (for small categories). The pasting scheme for connected wheel-free graphs (for properads) does _not_ satisfy this condition. We furthermore prove, assuming $\mathcal G$ is shrinkable and our base categories are nice enough, that a weak symmetric monoidal Quillen equivalence between two base categories induces a Quillen equivalence between their categories of $\mathcal G$-props. The final section gives illuminating examples that justify the conditions on base model categories.