It might be a dumb question but in a closed monoid category, shall we have a 'forgetful functor' sort of thing, from internal hom to the set of morphism? Or is it a part of definition?
@Mingcong: but note that unlike most "forgetful functors" there's no reason Hom(1, X) should be faithful; in other words, in general it forgets more than "structure," it forgets "stuff"
Interested beginners in algebraic topology should definitely try to get to the UChicago Summer school in Algebraic Topology this year: math.uchicago.edu/~chicagotopology2
