A PRO, an abbreviation for “product category”, is similar to a PROP but more general, being merely a monoidal category but not necessarily symmetric monoidal. They may be used to describe theories for finitary algebraic and coalgebraic structures which make sense in any monoidal category, for example monoids and comonoids.
A PRO is a strict monoidal category in which every object is of the form
for some . Somewhat more precisely, a PRO is a strict monoidal category for which the unique strict monoidal functor
is an isomorphism on objects.
Adam: Is there a variant where instead of a single object we start with some set of objects (or the objects of some other category) and require that any object of the new category is isomorphic to the tensor of zero or more objects from that set? I suppose this would be a “multi-sorted PRO” or something like that, although you’d have to specify the base set. Maybe “PRO generated by XYZ” or “PRO over XYZ” or some such thing. If there’s an established term for this, or something close to an established term, I’m keen on hearing about it.
Morphisms in a PRO are often thought of as operations which accept inputs and produce outputs, hence PROs are like nonpermutative operads but for the multiple outputs, which make them more general.
Let be a monoidal category. A -model in (or a -representation of ) is a strong monoidal functor ; the underlying object of the -model is the value . A -model is thus an object of equipped with operations
one for each morphism , reflecting the monoidal category structure of . A homomorphism of -models is a monoidal transformation between and ; it may be equivalently expressed as a morphism in which respects the modelings of the -operations .
The cube category is a PRO whose models are objects equipped with “elements” and a projection which is a retraction of both and , where denotes the monoidal unit.
The monoidal category of planar thickened 1d tangles (is this the right vocabulary?) is a PRO whose models are noncommutative Frobenius objects in monoidal categories.
More generally, given any doctrine , let be an initial algebra of the doctrine. A PRO-D can then be defined as an algebra such that the algebra map induces an isomorphism on objects. This is clearly an evil notion, and yet a useful and general one in practice.