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commutative monoid in a symmetric monoidal (infinity,1)-category

Definition

(…) symmetric monoidal (∞,1)-category (…)

$(\mathrm{\infty },1)$-Category of commutative monoids

Definition

For $C$ a symmetric monoidal (∞,1)-category write $\mathrm{CMon}(C)$ for the $(\mathrm{\infty },1)$-category of commutative monoids in $C$.

Properties

Theorem

• $\mathrm{CMon}(C)$ has all (∞,1)-coproducts and these are computed as tensor products in $C$.

• For $K$ a sifted (infinity,1)-category , (∞,1)-colimits of shape $K$ exist in $\mathrm{CMon}(C)$ and are computed in $C$ if $K$-colimits exist in $C$ are preserved by tensor product with any object.

• $\mathrm{CMon}(C)$ has all (∞,1)-limits and these are computed in $C$.

This is (Lurie, section 4).

Corollary

$(\mathrm{\infty },1)$-Colimits over simplicial diagrams exists in $\mathrm{CMon}(C)$ and are computed in $C$ if they exist in $C$ and a preserved by tensor products.

Because the simplex category is a sifted (infinity,1)-category (as discussed there).

References

• Jacob Lurie, Commutative algebra