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

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Models

Monoidal categories

monoidal categories

In higher category theory

Higher algebra

higher algebra

universal algebra

Contents

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

Definition

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

Properties

Theorem
• $CMon(C)$ 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 $CMon(C)$ and are computed in $C$ if $K$-colimits exist in $C$ are preserved by tensor product with any object.

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

This is (Lurie, section 4).

Corollary

$(\infty,1)$-Colimits over simplicial diagrams exists in $CMon(C)$ 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

Revised on August 12, 2014 18:12:28 by Urs Schreiber (24.213.171.170)