Relative left properness of colored operads

Abstract

The category of C-colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a relative left properness condition, ie that the class of weak equivalences between Σ-cofibrant operads is closed under cobase change along cofibrations. We also provide an example of Dwyer which shows that the model structure on C-colored symmetric operads is not left proper.

The homotopy theory of simplicial props

“…Here is an alternative suggestion for a method of proof, which we've only been able to work out parts of, and which presents considerable technical difficulties of its own. One would like to be able to adapt the proof of [3, 3.4] to the prop setting, but a major hurdle is that the category of props is not left proper (see [15]). On the other hand, we suspect that this category is 'relatively left proper' in the sense of [15], i.e.…”

“…One would like to be able to adapt the proof of [3, 3.4] to the prop setting, but a major hurdle is that the category of props is not left proper (see [15]). On the other hand, we suspect that this category is 'relatively left proper' in the sense of [15], i.e. that pushouts of weak equivalences between Σ-cofibrant props are again weak equivalences.…”

“…The model structure on sCat from Theorem 4.2 was right proper, and we now show that the same is true for the model structure on sProp. The category sProp is not left proper, due to a counterexample of Dwyer given in [15].…”

“…Focusing now on the transfusions q : (J, 2 ) → (J, 3 ) and f * q : (G, 2 ) → (G, f * 3 ), we have by (15)…”

“…hence we have that η(h) is the image under A of a morphism of id ×η(h) as in (32). Next consider q : (H , ) → (H , 0 ) from (30); by (15) we have that η(q) is the decomposition…”