nLab PRO

Idea

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.

Definitions

A PRO is a strict monoidal category $T$ in which every object is of the form

$$x^{\otimes n} = \underset{n \: times}{\underbrace{x \otimes \ldots \otimes x}} \qquad (product \: of \: n \: copies \: of \: x)$$

for some $n \geq 0$. Somewhat more precisely, a PRO is a strict monoidal category for which the unique strict monoidal functor

$$(\mathbb{N}, +, 0) \to (T, \otimes, I)$$

is an isomorphism on objects.

Adam: Is there a variant where instead of a single object $x$ 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.

RD : The term “coloured PROP” has been used. See arXiv:1207.2773

Morphisms $x^{\otimes m} \to x^{\otimes n}$ in a PRO are often thought of as operations which accept $m$ inputs and produce $n$ outputs, hence PROs are like nonpermutative operads but for the multiple outputs, which make them more general.

Let $C$ be a monoidal category. A $T$-model in $C$ (or a $C$-representation of $T$) is a strong monoidal functor $F: T \to C$; the underlying object of the $T$-model is the value $F(x)$. A $T$-model is thus an object $c$ of $C$ equipped with operations

$$\theta_c: c^{\otimes m} \to c^{\otimes n},$$

one for each morphism $x^{\otimes m} \to x^{\otimes n}$, reflecting the monoidal category structure of $T$. A homomorphism $F \to G$ of $T$-models $F, G: T \to C$ is a monoidal transformation between $F$ and $G$; it may be equivalently expressed as a morphism $\phi: F(x) \to G(x)$ in $C$ which respects the modelings of the $T$-operations $\theta$.

Examples

The augmented simplex category $\Delta_a$ is a PRO whose models are monoids in monoidal categories. Similarly, $\Delta_a^{op}$ is a PRO whose models are comonoids.

The cube category is a PRO whose models are objects $X$ equipped with “elements” $x_0, x_1: I \to X$ and a projection $p: X \to I$ which is a retraction of both $x_0$ and $x_1$, where $I$ 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.

Variations

Just as the notion of PROP is a symmetric monoidal analogue of PRO, there is a braided monoidal analogue called a PROB. The cartesian monoidal analogue is known as a Lawvere theory.

More generally, given any doctrine $D$, let $I(D)$ be an initial algebra of the doctrine. A PRO-D can then be defined as an algebra $T$ such that the algebra map $I(D) \to T$ induces an isomorphism on objects. This notion clearly violates the principle of equivalence, and yet is a useful and general one in practice.

References

An account of how the tensor product of PROs and the smash product of pointed spaces are two facets of the same construction is in

• Amar Hadzihasanovic, The smash product of monoidal theories, (arXiv:2101.10361)