nLab
PROP

Idea

A ‘PROP’ — an abbreviation of ‘products and permutations category’ — is a symmetric monoidal category generated by a single object, used to describe a given sort of algebraic structure. One can think of PROPs as a variant of Lawvere theories suitable for noncartesian contexts. In this respect they are similar to operads. However, they are more general, because they can be used to describe operations with many outputs as well as many inputs.

Definition

A PROP is a symmetric monoidal category where every object is of the form

$$x^{\otimes n}=x\otimes x\otimes \cdots \otimes x$$x^{\otimes n} = x \otimes x \otimes \cdots \otimes x

for a single object $x$ and $n\ge 0$.

Given a PROP $T$ and a symmetric monoidal category $C$, a symmetric monoidal functor

$$F:T\to C$$F : T \to C

is called an algebra or model of $T$ in $C$. The category of algebras of $T$ in $C$, say $\mathrm{Alg}(T,C)$, has

• symmetric monoidal functors $F:T\to C$ as objects,
• symmetric monoidal natural transformations as morphisms.

References

• Steve Lack, Composing PROPs, TAC 13 (2004), No. 9, 147–163.