When will a pointed category / arrow category be cartesian closed / a model category if the base category is cartesian closed / a model category?

For a category $\mathcal{C}$ with terminal object we have some construction on it :

• define the pointed category to be $*\downarrow \text{Id}$ the coslice category relative to the terminal object ;
• and we have the arrow category $\text{Id}\downarrow\text{Id}$ ;
• furthermore, we define the category of pairs to be the subcategory of the arrow category where the objects are not morphism pairs, but monomorphism pairs (or regular monomorphism?).

If $\mathcal{C}$ is cartesian closed, under what condition will the three previous categories be cartesian closed with smash product and map of pairs? Same question if $\mathcal{C}$ is a model category. Any reference?

For $f\colon A\to X$ and $g\colon B\to Y$,

• smash product:
• pushout of $\mathrm{id}\times f\colon A\times B \to A\times Y$ and $g\times \mathrm{id}\colon A\times B\to X\times B$ denoted by $(X, A)\otimes (Y, B)$
• the map $(X, A)\otimes (Y, B)\to X\times Y$
• map of pairs:
• because the functor $(-)^A$ is continuous, we have $g'\colon B^A\to Y^A$
• composing $\mathrm{id}\times f\colon Y^X\times A\to Y^X\times X$ and the evaluation map $\epsilon\colon Y^X\times X\to Y$ we have a function $Y^X\times A\to Y$ then we have induced a function $f'\colon Y^X\to Y^A$
• we have the pullback of the two functions $f'$ and $g'$ denoted by $(Y, B)^{(X, A)}$
• the map $B^X\to (Y, B)^{(X, A)}$

We know the cartesian closed question is true for pointed category of compactly generated weak Hausdorff space.

If $\mathcal{C}$ is a model category we know the slice category and coslice category is a model category.

(These are results from Peter May's Concise Algebraic Topology and More Concise Algebraic Topology.)

To what extend can we generalise these results?

Here are the answers, in no particular order:

• Categories with a zero object are cartesian closed if and only if they are trivial – see e.g. here. In particular, the category of pointed objects is almost never cartesian closed.
• The smash product is not the cartesian product in the category of pointed objects. However, under good conditions, it is a symmetric monoidal product and we get a symmetric monoidal closed category.
• The category of arrows is cartesian closed if the original category is cartesian closed and has finite limits.
• The category of arrows has a Reedy model structure if the original category has a model structure – see e.g. [Hovey, Model categories, §5.2] or [Hirschhorn, Model categories and their localizations, Ch. 15].
• Under good conditions, the category of pairs will be a reflective subcategory of the category of arrows; and if the reflector preserves finite products, then the category of pairs will be cartesian closed (if the category of arrows is cartesian closed).

I don't know if the category of pairs inherits a model structure. Sometimes model structures can be transferred across adjunctions, sometimes not.

• any reference about "under good conditions" for pointed category? – Minghao Liu May 17 '14 at 12:20
• I think it suffices to have a cartesian closed category with equalisers and reflexive coequalisers. See the discussion here. – Zhen Lin May 17 '14 at 12:38
• ... and also finite coproducts, of course. – Zhen Lin May 17 '14 at 12:45
• A very general consideration of the smash products appears in my thesis (which I can link here in a couple of weeks, perhaps). – Martin Brandenburg May 18 '14 at 7:44
• @MartinBrandenburg Please do post the link here when available, thanks! – Minghao Liu May 19 '14 at 3:52