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Have added to pushout-product the statement (here) that pushout product $\Box$ of $I_1$-cofibrations with $I_2$-cofibrations lands in $(I_1\Box I_2)$-cofibrations; and (here) the example of pushout products of the inclusions $S^{n-1} \hookrightarrow D^n$. Both without proof for the moment.
Here is a terminology question. Does anyone know or have suggestions for a nice name for the right adjoint(s) to the pushout product? Whenever I have to name one, I usually end up saying something like “dual pushout product” which is a mouthful and difficult to work into sentences. As explained in Joyal-Tierney calculus they are sometimes called “left quotient” and “right quotient” which I don’t like, these constructions are nothing like quotients. (I can only imagine that this arose because when the tensor is join of simplicial sets, then the right adjoints are slices and they are sometimes notated like quotients. But that’s just a coincidence of notation.) I was also thinking about directly dualizing “pushout product” to something like “pullback hom” which is sort of descriptive but just doesn’t sound good to me.
Can anyone help me with this riddle and suggest something that is both evocative and reasonably concise?
I have been saying “pullback powering” for it.
Thanks for a suggestion. “Pullback power” is similar to my “pullback hom”, but I agree that it sounds less awkward. That’s probably a good name.
That is pretty good. Is it recorded on the nLab anywhere?
I have used that term without much ado in entries such as enriched model category, classical model structure on topological spaces, model structure on orthogonal spectra.
Ok, I created a stub pullback power.
I actually used “pullback hom” in my papers.
It may sound awkward, but it seems more descriptive than “pullback power”.
The thing is that while right adjoints to some tensors are really hom objects, sometimes they are more general. I think “pullback power” is sufficiently generic to describe all such situations. Also “power” sounds to me more like an actual noun than “hom”.
In A Generalized Blakers-Massey Theorem they call the dual of the pushout product, ’pullback hom’ (3.1.2).
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