nLab
cocartesian monoidal (infinity,1)-category

Contents

Context

(,1)(\infty,1)-Category theory

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

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

(…)

(Lurie, def. 2.4.0.1)

(…)

Properties

Lemma

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

Theorem

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

See (Lurie, Theorme 2.4.3.18).

Examples

References

Section 2.4 of

Created on March 6, 2015 at 18:05:52. See the history of this page for a list of all contributions to it.