Re: Closed symmetric monoidal category

Re: Closed symmetric monoidal category

[george.janelidze]

Dear Johannes,

I don't know how much it was studied, but let me mention, just in case, the
well-known Proposition 18.3 of

S. Mac Lane, Categorical algebra, Bull. Amer. Math. Soc. 71 (1965), 40–106,

even though there is no "closed" mentioned there (also there is no
"symmetric", but defining tensored category in that paper, Mac Lane also
means "symmetric"). It says:

If D is a tensored category, so is the category of modules over a
commutative D-algebra A

(I wrote A instead of Greek Lambda)

- and the fact (also well-known of course) that A could be a differential
graded commutative algebra is visible in Section 17 (Take D = DG(K-Mod) in
Mac Lane's notation).

Best regards, George



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Closed symmetric monoidal category

[Johannes Huebschmann]

Dear All

The category of modules over a differential graded
commutative algebra A,
with the tensor product over A
(suitably interpreted) as operation of composition,
the direct sum as operation of biproduct,
the algebra A as identity object,
and the interchange map (suitably interpreted) as
operation of braiding,
  is a closed symmetric monoidal
category (unless I am mistaken).

Is there a place in the literature where
this category is studied?

Best regards

Johannes



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