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Alexander Gietelink Oldenziel (May 15 2024 at 12:58):I'd like to understand how enriched (co)products in enriched category are defined and concretely computed in examples.
I have found Kelly's canonical text a little heavy going.
I am particularly interested in the example of Vect enriched categories.
In this case, I wonder how the definition of an enriched product shakes out.
Here are three guesses
(1) the enriched product coincides with the ordinary (bi)product in Vect enriched categories.
'something something conical limits
(2) For a Vect-category $C$ the enriched product of $X,Y \in C$ is given by an object $X\times Y$ in $C$ such that the Vect-presheaf $F: C^{op} \to Vect$ given by $Z \mapsto Hom(Z,X)\otimes_k Hom(Z,Y)$ is represented by the object $X\times Y$.
(3) the enriched products is something entirely more mysterious.
Mike Shulman (May 15 2024 at 15:24):(2) is correct if you replace with , and then it is true for arbitrary enriching category. (1) is correct for Vect because the forgetful functor from Vect to Set is conservative; in general, an enriched product is an ordinary product but the converse may not be true.
Mike Shulman (May 15 2024 at 15:25):(You have to put double $$s around math here.)