I first asked this on StackExchange, but no dice; so apologies in advance if this question really belongs there.


Suppose a functor $F \colon \mathcal{C} \to \mathcal{D}$ between two model categories (i) sends cofibrant objects to cofibrant objects and (ii) sends weak equivalences between cofibrant objects to weak equivalences (between cofibrant objects).


Is there a standard name for this kind of functor? It's not necessarily left Quillen, so calling it "left homotopical" seems reasonable, but I wasn't able to find a reference in the literature. Thanks in advance!



Terminology: "left homotopical"?
Thanks to Zhen Lin and Denis-Charles Cisinski for their comments, as well as "jgon" who commented on the Stack Exchange post. It seems there is no standard terminology for such a functor.


There is, however, a related notion. A functor that preserves weak equivalences between cofibrant objects is said to be *left deformable* in the language of either DHKS "Homotopy Limit Functors on Model Categories and Homotopical Categories" or Blumberg-Riehl "Homotopical resolutions associated to deformable adjunction's." A left deformable functor, however, need not send cofibrant objects to cofibrant objects.


