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In
mathematics, a '''strict monoidal category''' is a
category with a product operation × on
objects that has properties analogous to those of the
tensor product. The product is assumed to be
associative, and have an
left and right identity, I. The corresponding morphisms, expressing the 'associative law', and 'identity element' properties, are subject to a system of
coherence conditions: these are familiar in the cases of the examples given below, but in general require some small amount of
syntax to express. The concept of '''braided monoidal category''' has been much studied from the 1980s onwards; it occurs in
string theory applications, and is a more 'relaxed' theory defined by fewer such coherence conditions.
Any category with standard
categorical products and a
terminal object is a strict monoidal category, with the categorical product as tensor product and the terminal object as identity. Also, any category with
coproducts and an
initial object is a strict monoidal category - with the coproduct as tensor product and the initial object as identity. However, in many monoidal categories (such as
K-Vect, given below) the tensor product is neither a categorical product nor a coproduct.
Examples of monoidal categories, illustrating the parallelism between the category of vector spaces over a field and the category of sets, are given below.
| K-Vect | Set |
|---|
| Given a field (or commutative ring) K, the category K-Vect is a symmetric monoidal category with product ⊗ and identity K. |
The category Set is a symmetric monoidal category with product × and identity {*}. |
| A unital associative algebra is an object of K-Vect together with morphisms <math>\nabla:A\otimes A\rightarrow A</math> and <math>\eta:\mathbb{K}\rightarrow A</math> satisfying Image:Algebra.png |
A monoid is an object M together with morphisms <math>\circ:M\times M\rightarrow M</math> and <math>1:{*}\rightarrow M</math> satisfying Image:Monoid.png. |
| A coalgebra is an object B with morphisms <math>\Delta:B\rightarrow B\otimes B</math> and <math>\epsilon:B\rightarrow\mathbb{K}</math> satisfying Image:Coalg.png. |
Any object of Set, S has two unique morphisms <math>\Delta :S\rightarrow S\times S</math> and <math>\epsilon:S\rightarrow\{*\}</math> satisfying Image:Diag.png. In particular, ε is unique because {*} is a terminal object. |
