# Shrinkability, relative left properness, and derived base change

Research paper by Philip Hackney, Marcy Robertson, Donald Yau

Indexed on: 05 Oct '15Published on: 05 Oct '15Published in: Mathematics - Algebraic Topology

#### 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 for weak equivalences between equivariantly-cofibrant $\mathcal G$-props. Connected pasting schemes satisfying this closure property include those for all connected wheeled graphs (for wheeled properads), wheeled trees (for wheeled operads), simply connected graphs (for dioperads), half-graphs (for half-props), 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, that a weak symmetric monoidal Quillen equivalence between two base categories induces a Quillen equivalence between their categories of $\mathcal G$-props.