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Created a little entry Vect(X) (to go along with Vect) and used the occasion to give distributive monoidal category the Examples-section that it was missing and similarly touched the Examples-section at rig category.
Oops, that comment was supposed to go on the vector bundles page.
I see that “VectBund” was redirecting to here, while it ought to be pointing to an entry about the category of vector bundles over arbitrary base spaces, with morphisms covering non-trivial maps of base spaces.
I am removing the redirect now and will instead start an entry VectBund.
Assuming X is a smooth manifold, has Vect(X) been given a smooth tensor category structure? I understand one would exhibit Vect(X) as a sheaf on say CartSp with the tensor operations but has this been done explicitly anywhere?
Let’s see. With Vect denoting the 2-sheaf of categories of vector bundles over Cartesian spaces, you probably mean by a smooth form of Vect(X) the internal hom [X,Vect] in the 2-topos over CartSp. Now the CartSp-wise tensor product of vector bundles should make Vect a monoid object in that 2-topos, I suppose, whence the question comes down to whether the internal hom into a monoid object inherits monoid structure — which is clearly the case, I’d think.
I see, thanks, Urs.
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