nLab strict monoidal category

Contents

Context

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

Definition

A monoidal category is strict if its associator and left/right unitors are identity natural transformations.

Very explicitely, it means that:

Definition

A strict monoidal category is a category 𝒞\mathcal{C} equipped with an object 1𝒞1 \in \mathcal{C} and a bifunctor :𝒞×𝒞𝒞\otimes:\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} such that for every objects A,B,CA,B,C and morphisms f,g,hf,g,h, we have:

  • (AB)C=A(BC)(A \otimes B) \otimes C = A \otimes (B \otimes C)
  • 1A=A1 \otimes A = A
  • A1=AA \otimes 1 = A
  • (fg)h=f(gh)(f \otimes g) \otimes h = f \otimes (g \otimes h)
  • Id 1f=fId_{1} \otimes f = f
  • fId 1=ff \otimes Id_{1} = f

Last revised on August 1, 2022 at 14:09:28. See the history of this page for a list of all contributions to it.