nLab cocartesian monoidal (infinity,1)-category

Idea

A cocartesian monoidal (∞,1)-category is a symmetric monoidal (∞,1)-category whose tensor product is given by the categorical coproduct. This is dual to the notion of cartesian monoidal (∞,1)-category.

Definition

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(Lurie, def. 2.4.0.1)

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Properties

Lemma

If $C$ is an (infinity,1)-category admitting finite coproducts, then the cocartesian symmetric monoidal structure on $C$ is equivalent to the symmetric monoidal structure obtained as the opposite of the cartesian monoidal structure on the opposite (infinity,1)-category $C^{op}$.

Theorem

If $C$ is an (infinity,1)-category with finite coproducts, then the cocartesian model structure on $C$ is universal among symmetric monoidal (infinity,1)-categories $D$ for which there exists a functor $C \to CAlg(D)$, to the symmetric monoidal (infinity,1)-category of commutative algebra objects in $D$.

See (Lurie, Theorme 2.4.3.18).

Examples

• If $C$ is a symmetric monoidal (infinity,1)-category, then the canonical symmetric monoidal structure on $CAlg(C)$, the (infinity,1)-category of commutative algebra objects in $C$, is cocartesian, by (Lurie, Proposition 3.2.4.7).

References

Section 2.4 of

• Jacob Lurie, Higher Algebra