# nLab commutative algebra in an (infinity,1)-category

higher algebra

universal algebra

# Idea

Recall the pattern in ordinary algebra:

Accordingly in higher algebra:

# Definition

A commutative algebra object in a symmetric monoidal (infinity,1)-category $C$ is a lax symmetric monoidal $\left(\infty ,1\right)$-functor

$*\to C\phantom{\rule{thinmathspace}{0ex}}.$* \to C \,.

In more detail, this means the following:

###### Definition

Given a symmetric monoidal (infinity,1)-category in its quasi-categorical incarnation as a coCartesian fibration of simplicial sets

$p:{C}^{\otimes }\to N\left({\mathrm{FinSet}}_{*}\right)$p : C^\otimes \to N(FinSet_*)

a commutative algebra object in $C$ is a section

$A:N\left({\mathrm{FinSet}}_{*}\right)\to {C}^{\otimes }$A : N(FinSet_*) \to C^\otimes

such that $A$ carries collapsing morphisms in ${\mathrm{FinSet}}_{*}$ to coCartesian morphisms in ${C}^{\otimes }$.

# References

the above definition is definition 1.19 in

Revised on September 15, 2009 15:39:41 by Urs Schreiber (195.37.209.182)