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

Question

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?

Answer 1

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.