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Definition of enriched caterories or internal homs without using monoidal categories.

This is exactly the notion of a closed category. See Eilenberg and Kelly's article in the 1965 La Jolla proceedings (Springer 1966). I think they also describe categories enriched in a closed catego …
Finn Lawler's user avatar
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A better way to compute the mapping spaces of the category of spans in an enriched tensored ...

Not quite an answer, but I hope it helps: Your second definition looks (somewhat) like the usual definition of the V-valued hom of V-functors $[Sp,X] (a,b) = \int_{A \in Sp} X(aA, bA)$. If that's rig …
Finn Lawler's user avatar
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