nLab cartesian monoidal (infinity,1)-category

Idea

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

In the special case that the underlying (∞,1)-category is equivalent to just a 1-category, then this is equivalently a cartesian monoidal category.

Definition

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(Lurie, def. 2.4.0.1)

(…)

Properties

Coalgebra objects

Every object in a Cartesian monoidal $\infty$-category is canonically a comonoid object via the diagonal map. See also at (infinity,n)-category of correspondences the section Via coalgebras.

References

Section 2.4 of

• Jacob Lurie, Higher Algebra