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Stream: learning: questions

Topic: def Quillen bifunctor

Daniel Plácido (Jun 15 2021 at 13:36):

The definition of Quillen bifunctor at nLab (and everywhere else) isn't the most natural one - which to me would be a bifunctor $F:C\times D\to E$, where $C,D,E$ are model categories, such that $F(c,-)$ and $F(-,d)$ are left or right Quillen functors in each entrance. Is this idea anyhow equivalent to the actual definition? If not, why is the definition so clouded?

Zhen Lin Low (Jun 15 2021 at 14:29):

No. The definition is constructed so that the motivating examples satisfy the definition!

Mike Shulman (Jun 15 2021 at 14:38):

And so as to make it more useful.

Daniel Plácido (Jun 15 2021 at 20:41):

ok, makes sense, thanks! I was wondering because iirc $X\times (-):\mathsf{Top}\to\mathsf{Top}$ is left Quillen

Reid Barton (Jun 15 2021 at 21:48):

It's not left Quillen unless $X$ is cofibrant, because a left Quillen functor has to preserve cofibrant objects but $X \times * = X$. (I'm assuming you're using the standard model structure on Top, in which not every object is cofibrant.)