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. They also generalize properads and dioperads?.
A PROP is a symmetric monoidal category where every object is of the form
for a single object and .
A colored PROP with set of colors is a symmetric strict monoidal category whose monoid of objects is freely generated by .
A morphism of PROPs is just a functor of symmetric monoidal categories that takes the generators for the objects of to the generators of the objects of . Equivalently, a morphism of PROPs is a pair of functions: one from the colors of to the colors of , , and for each pair of finite list of colors and in , a function . The two functions and are of course required to preserve compositions, units and symmetries.
Thus there is a category of PROPs. For more on this category, as well as some its properties, see HR1.
Given a PROP and a symmetric monoidal category , a symmetric monoidal functor
is called an algebra or model of in . The category of algebras of in , say , has
Note that all of the above definitions can be enriched over a symmetric monoidal category which yields the notion of enriched PROPs. For instance, we can have simplicial and topological PROPs where the sets of morphisms are simplicial sets or topological spaces.
First, in the case that we are working only with simplicial PROPs with a fixed set of colors (or, in other words, PROPs whose free monoids of objects all have the same generators), we have the following theorem of HR2:
There is a cofibrantly generated model structure on the category of simplicial PROPs with a fixed set of colors in which a morphism of simplicial PROPs is a weak equivalence (resp. fibration) if for each simplicial set of morphisms the induced map is also a weak equivalence (resp. fibration).
There is a cofibrantly generated model structure on the category of all simplicial PROPs where a morphism of PROPs is a weak equivalence (resp. fibration) if:
Note that the model structure on simplicial PROPs is not the model structure on gets by lifting the model structure of simplicial operads along the free forgetful adjunction between simplicial operads and simplicial PROPs.
A perhaps paradigmatic example is that there is a -enriched prop whose algebras are bialgebras. It should be observed here that there is no -enriched operad (or cooperad) whose algebras are bialgebras, so this is a genuine example that illustrates a gain in generality of props over operads.
See Pirashvili for some more details on this prop.
Every PROP defines a polycategory; see there for more. Note that PROPs are strictly more general than polycategories since in a PROP we can compose along many objects at once. This restriction actually makes polycategories more like dioperads? than PROPs (cf. Gan).
Teimuraz Pirashvili, On the PROP corresponding to bialgebras, http://arxiv.org/abs/math/0110014. (link)