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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?

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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.

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  • $\begingroup$ any reference about "under good conditions" for pointed category? $\endgroup$ Commented May 17, 2014 at 12:20
  • $\begingroup$ I think it suffices to have a cartesian closed category with equalisers and reflexive coequalisers. See the discussion here. $\endgroup$
    – Zhen Lin
    Commented May 17, 2014 at 12:38
  • $\begingroup$ ... and also finite coproducts, of course. $\endgroup$
    – Zhen Lin
    Commented May 17, 2014 at 12:45
  • $\begingroup$ A very general consideration of the smash products appears in my thesis (which I can link here in a couple of weeks, perhaps). $\endgroup$ Commented May 18, 2014 at 7:44
  • $\begingroup$ @MartinBrandenburg Please do post the link here when available, thanks! $\endgroup$ Commented May 19, 2014 at 3:52

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