General construction for internal hom in a presheaf category - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:37:50Z http://mathoverflow.net/feeds/question/10406 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10406/general-construction-for-internal-hom-in-a-presheaf-category General construction for internal hom in a presheaf category Harry Gindi 2010-01-01T21:17:44Z 2010-01-01T22:22:23Z <p>I was reading about the internal hom functor for simplicial sets, and the construction is very "localized" (nothing to do with localization, just the english word). It seems like there should be a general construction for any presheaf category that would be similar to this. That is, an actual construction, not just the existence of the functor provided by the theorem that every Grothendieck topos is a Lawvere topos and therefore cartesian closed. Does such a construction exist, and if so, can you give a reference?</p> http://mathoverflow.net/questions/10406/general-construction-for-internal-hom-in-a-presheaf-category/10410#10410 Answer by Mike Shulman for General construction for internal hom in a presheaf category Mike Shulman 2010-01-01T22:22:23Z 2010-01-01T22:22:23Z <p>The formula for the internal hom between presheaves $F\colon C^{op}\to Set$ and $G\colon C^{op}\to Set$ can be derived from the Yoneda lemma. Given $c\in C$, we know that we must have $G^F(c) \cong Hom(y(c), G^F) \cong Hom(y(c) \times F, G)$ so we can simply define $G^F(c) = Hom(y(c) \times F, G)$, which is evidently a presheaf on $C$. The isomorphism $Hom(H,G^F)\cong Hom(H\times F, G)$ for non-representable $H$ then follows from the fact that every presheaf $H$ is canonically a colimit of representables, and $Hom(-,G^F)$ and $Hom(-\times F,G)$ both preserve colimits (the former by definition of colimits, and the latter by that and since limits and colimits in presheaf categories are computed pointwise and products in $Set$ preserve colimits).</p> <p>This is Proposition I.6.1 in "Sheaves in geometry and logic."</p>