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Stream: learning: questions

Topic: reference: enriched cartesian closure

Todd Trimble (Dec 29 2021 at 03:25):

(I have no idea which stream or category best fits the question; please feel free to move it if you know better.)

If $V$ is a complete, cocomplete, cartesian closed category, and $C$ is a small $V$-category, then $V^{C^{op}}$ is cartesian closed in the enriched sense. I know very well how to prove this, so I'm not asking for help with proving it, but can someone point me to a place in the literature where this is stated and proved? For a paper I'm involved in, it would be nice not to have to go to the bother of proving it from scratch.

Todd Trimble (Dec 29 2021 at 15:52):

Ah... for what interest this may have, I think I found a good reference: Day, On closed categories of functors. That's where promonoidal structures (what he called then premonoidal structures) first appeared; what I hadn't appreciated until now is that in this context, we can define for any $V$-category $C$ a promonoidal structure

$C^{op} \times C^{op} \times C \stackrel{1 \times 1 \times \delta}{\to} C^{op} \times C^{op} \times C \times C \stackrel{\hom_{C \times C}}{\to} V$

so that the statement of my previous post is a special case that falls out from the general facts about promonoidal structures (see for example Day's Example 5.2, page 36). So I'm satisfied.