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I added an explicit definition of cartesian model category to cartesian closed model category to highlight the convention that the terminal object is assumed cofibrant.
I have added to cartesian model structure and to model structure on topological spaces the remark that the standard model structure on compactly generated weakly Hausdorff topological spaces is a cartesian model structure
Re #3:
Removed “closed” from the title.
I was feeling we should, in contrast, expand out the title to “cartesian closed monoidal category”.
Because the word “cartesian category” is ambiguous, and yet “cartesian closed monoidal category” is not among its usual uses.
In any case, I made “cartesian closed monoidal model category” a redirect here.
Re #5: My use of “cartesian model category” was based on the fact that this particular term is commonly used in existing literature, e.g., in Simpson’s book.
Whereas cartesian category is indeed ambiguous, cartesian model category is not, because one of the meanings (category with finite products) becomes vacuous for model categories (which always have finite products by definition).
It’s pretty standard to say monoidal model category without adding the adjective “closed”, so it seems reasonable to me to omit it here too. I would probably lean towards “cartesian monoidal model category” myself, with a remark and redirect for the common shortening “cartesian model category”.
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