nLab cartesian monoidal (infinity,1)-category

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Contents

Context

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

Monoidal categories

monoidal categories

With braiding

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 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 𝒞 \mathcal{C}^{\otimes} is Cartesian if the following conditions are satisfied:

  1. The unit object 1 𝒞𝒞1_{\mathcal{C}} \in \mathcal{C} is final, and

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

    CC1 𝒞CD1 𝒞DD C \simeq C \otimes 1_{\mathcal{C}} \leftarrow C \otimes D \rightarrow 1_{\mathcal{C}} \otimes D \simeq D

    exhibit CDC \otimes D as the product C×DC \times D.

Equivalently, for n0n \geq 0, we stipulate that the nn-ary tensor product :𝒞 n𝒞\otimes:\mathcal{C}^n \rightarrow \mathcal{C} is right adjoint to the nn-ary diagonal map Δ:𝒞𝒞 n\Delta:\mathcal{C} \rightarrow \mathcal{C}^n.

Properties

Rigidity

HA 2.4.1.5 constructs a Cartesian symmetric monoidal ∞-category structure 𝒞 ×\mathcal{C}^\times on 𝒞\mathcal{C} whenever 𝒞\mathcal{C} has all finite products, and by unwinding definitions, 𝒞\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 histhe particular construction of HA 2.4.1.5, including constructing a canonical equivalence. We may summarize these results as the following, wherein Cat ×\mathrm{Cat}_\infty^\times denotes the subcategory of Cat \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 () ×:Cat ×Cat (-)^\times:\mathrm{Cat}_\infty^{\times} \hookrightarrow \mathrm{Cat}_\infty^{\otimes} whose image is spanned by the Cartesian symmetric monoidal \infty-categories.

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

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

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

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

Proposition

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

References

Section 2.4 of

Last revised on August 9, 2024 at 12:27:59. See the history of this page for a list of all contributions to it.