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Consider a category C with weak equivalences, e.g., a model category.

For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) along f is also a homotopy pushout.

This notion is closely related to that of h-cofibration (alias flat map, cosharp map, left proper map, W-cofibration, cofibrillation, weak cofibration, see the terminological discussion at n-Café). The latter is defined as a morphism f: A→B for which the induced cobase change functor f_*: A/C→B/C preserves weak equivalences.

If C is a left proper model category, then i-cofibrations coincide with h-cofibrations and contain cofibrations, see, for example, Lemma 1.2 and Proposition 1.5.(i,ii) in Batanin and Berger's “Homotopy theory for algebras over polynomial monads” (arXiv:1305.0086).

If C is not left proper, then i-cofibrations do not necessarily coincide with h-cofibrations.

Has the notion of i-cofibration been studied before? Does it have a name of its own? Any references on this matter will be appreciated.

Motivation for this question comes from a desire to study (homotopy) pushouts in model categories of algebras over operads, which are rarely left proper.

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  • $\begingroup$ The same motivation applies to calculating homotopy pushouts of operads themselves. $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Sep 25, 2014 at 18:13
  • $\begingroup$ @GabrielC.Drummond-Cole: Of course! The category of operads itself is the category of algebras over a certain colored operad (the “operad of operads”), so this example is covered by the main post. The example that I have in mind is that of enriched categories in enriched categories, i.e., higher enriched categories. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 26, 2014 at 10:19

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