abstract nonsense 

wait wtf I always thought currying, in like, the CS sense, and the tensor-hom adjunction for modules or were basically the same thing. so I was going around thinking "oh yeah that means R-modules, R-algebras, etc. are all cartesian closed, cartesian closed just means there's an internal hom"

but that's not true, since the tensor product of modules is *not* the categorical product. it's even less true for algebras, where the tensor product is the *co-product* of all things. this is fucking with me, because these cases are *the* example of "internal hom" in my head

in either of these cases, does the product actually have a right adjoint? I don't think so... In the case of vector spaces, definitely not, since the dimension would be fucky...


abstract nonsense 

that sounds like it would mean R-alg is "cocartesian closed", but according to ncatlab.org/nlab/show/cocartes that can't possibly be right...

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abstract nonsense 

ahh I need to actually feed myself instead of trying to read nLab

abstract nonsense 

oh nevermind I forgot for algebras Hom(A,B) is just a module, not an algebra. crisis averted.

still pretty weird though

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