Definition of enriched caterories or internal homs without using monoidal categories. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:33:10Z http://mathoverflow.net/feeds/question/79102 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79102/definition-of-enriched-caterories-or-internal-homs-without-using-monoidal-categor Definition of enriched caterories or internal homs without using monoidal categories. Garlef Wegart 2011-10-25T18:43:07Z 2011-10-25T18:50:07Z <p>I know this question may seem nonsensical at first but let me exlain what i have in mind:</p> <p>In enriched category theory we define categories enriched over a monoidal category $(\mathcal{V},\otimes, I)$. An enriched category then is given by a set/class of objects $\mathcal C$ and a rule assigning to every pair $X,Y$ of such objects a hom-object $[X,Y]$. Furthermore we define composition and identities using $\otimes$ and $I$, remodelling the definitions of usual category theory.</p> <p><strong>Now for the question</strong>: Can we go the other way around?</p> <p>Let's stick to internal homs for the beginning: Given a category $\mathcal V$ ; can say what additional data turn a functor $$[-,-]:\mathcal{V}^{\mathrm{op}}\times\mathcal V\to \mathcal V$$ into something like an internal hom?</p> <p>In the case of $[X,-]$ having a left adjoint $-\otimes X$ for every $X$, these additional data should result in $(\mathcal V,\otimes)$ becomming a closed monoidal category with internal hom isomorphic to $[-,-]$.</p> http://mathoverflow.net/questions/79102/definition-of-enriched-caterories-or-internal-homs-without-using-monoidal-categor/79103#79103 Answer by Finn Lawler for Definition of enriched caterories or internal homs without using monoidal categories. Finn Lawler 2011-10-25T18:50:07Z 2011-10-25T18:50:07Z <p>This is exactly the notion of a <a href="http://en.wikipedia.org/wiki/Closed_category" rel="nofollow">closed category</a>. See Eilenberg and Kelly's article in the 1965 La Jolla proceedings (Springer 1966). I think they also describe categories enriched in a closed category.</p>