Re: Are sheaves a symmetric monoidal closed category?


Jannick Asmus ha escrit:

Please let me know whether this is helpful for you.



Indeed it was!

I think you have to forget to talk about elements of objects of C
like [F(V), G(V)] , but, if I'm not wrong this doesn't matter because
one can write your idea in the following "element-free" way.

First, you have a bifunctor

Open(X) x Open(X)^op ---> C

where Open(X) is the category of open sets of X , defined on
objects by

( V , V' ) |---> [F(V), G(V')] .

Assuming that C is complete, this bifunctor has an end, which is a
functor

Open(X)^op ---> C

and is exactly your U |---> P(U) .

Moreover, since an end is a limit it commutes with other limits. So
this presheaf is in fact a sheaf!

As for the fact of being adjoint to the tensor product, it seems to me
that, playing a little with ends, it works.

Thank you very much!


Agust? Roig
.

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