nLab cartesian monoidal (infinity,1)-category

Idea

A cartesian symmetric 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 symmetric 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 symmetric monoidal category?.

Definition

Definition

A symmetric monoidal ∞-category $\mathcal{C}^{\otimes}$ is Cartesian if the following conditions are satisfied:

1. The unit object $1_{\mathcal{C}} \in \mathcal{C}$ is final, and

2. For every pair of objects $C,D \in \mathcal{C}$, the canonical maps

   $$C \simeq C \otimes 1_{\mathcal{C}} \leftarrow C \otimes D \rightarrow 1_{\mathcal{C}} \otimes D \simeq D$$

   exhibit $C \otimes D$ as the product $C \times D$.

Equivalently, for $n \geq 0$, we stipulate that the $n$-ary tensor product $\otimes:\mathcal{C}^n \rightarrow \mathcal{C}$ is right adjoint to the $n$-ary diagonal map $\Delta:\mathcal{C} \rightarrow \mathcal{C}^n$.

Properties

Rigidity

HA 2.4.1.5 constructs a Cartesian symmetric monoidal ∞-category structure $\mathcal{C}^\times$ on $\mathcal{C}$ whenever $\mathcal{C}$ has all finite products, and by unwinding definitions, $\mathcal{C}$ has all finite products if it attains a Cartesian symmetric monoidal structure.

Additionally, HA 2.4.1.7 characterizes functors from Cartesian symmetric monoidal $\infty$-categories to the particular construction of HA 2.4.1.5, including constructing a canonical equivalence. We may summarize these results as the following, wherein $\mathrm{Cat}_\infty^\times$ denotes the subcategory of $\mathrm{Cat}_\infty$ whose objects are $\infty$-categories possessing finite products and whose morphisms are product-preserving functors.

Theorem

Lurie’s construction yields a fully faithful functor $(-)^\times:\mathrm{Cat}_\infty^{\times} \hookrightarrow \mathrm{Cat}_\infty^{\otimes}$ whose image is spanned by the Cartesian symmetric monoidal $\infty$-categories.

$\mathcal{O}$-algebras as $\mathcal{O}$-monoids

Given $\mathcal{O}^{\otimes}$ an ∞-operad and $\mathcal{C}$ an ∞-category, we say that a functor $\mathcal{O}^{\otimes} \rightarrow \mathcal{C}$ is an $\mathcal{O}$-monoid if, given an object $\mathbf{X} = (X_i \in \mathcal{O})_{i \leq n} \in \mathcal{O}_{\langle n \rangle}$ the canonical maps

$$\left\{M(\mathbf{X}) \rightarrow M(X_i) \right\}$$

induced by cocartesian transport in $\mathcal{O}^\otimes$ exhibit $M(\mathbf{X})$ as a product $M(\mathbf{X}) \simeq \prod_{i \leq n}(M(x_i))$.The following is HA 2.4.2.5.

Proposition

If $\mathcal{C}$ has finite products, then the forgetful functor $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C}^{\times}) \rightarrow \mathrm{Fun}(\mathcal{O}^{\otimes}, \mathcal{C})$ is fully faithful with image spanned by the $\mathcal{O}$-monoids.

Coalgebra objects

Every object in a Cartesian monoidal $\infty$-category is canonically a comonoid object via the diagonal map (just as in the 1-categorical case here).

See also at (infinity,n)-category of correspondences the section Via coalgebras.]

Related concepts

• cocartesian monoidal (infinity,1)-category

References

Section 2.4 of

• Jacob Lurie, Higher Algebra