Monoidal category

Strict monoidal category

A strict monoidal category is a monoidal category in which the associator 𝛼, left-unitor 𝜆, and right-unitor 𝜌 are all the identity natural transformation. cat Thus a strict monoidal category is precisely a Monoid object in Category of small categories. Explicitly, a strict monoidal category 𝖢 is equipped with

1. a functor ( ⊗) :𝖢 ×𝖢 →𝖢 called the tensor product; and
2. an object 𝟙 ∈𝖢 called the tensor unit

such that

𝑥⊗𝟙=𝑥=𝟙⊗𝑥

for any object 𝑥 ∈𝖢 and

(𝑥⊗𝑦)⊗𝑧=𝑥⊗(𝑦⊗𝑧)

for any objects 𝑥,𝑦,𝑧 ∈𝖢.

Examples

TABLE without id
  ("[[" + file.path + "|" + categoryName + "]]") as name,
  default(symbol, mathLink) as symbol,
  object,
  morphism,
  tensorProduct as product,
  tensorUnit as unit,
  arguments
FROM #monoidal-category/strict

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