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# Contents

## 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. Because of their one-sorted nature, 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

###### Definition

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

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

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

There is also a notion of colored PROP akin to colored operads, or multisorted Lawvere theories. One way to define a colored PROP is as a certain kind of symmetric monoidal category (see Remark 2.2.14 of Yau):

###### Definition

A colored PROP with set of colors $\mathfrak{C}$ is a strict symmetric monoidal category $(P,\odot)$ whose monoid of objects is freely generated by $\mathfrak{C}$.

###### Definition

A morphism of PROPs $\phi:Q\to T$ is just a strict symmetric monoidal functor that takes the generators for the objects of $Q$ to the generators of the objects of $T$. Equivalently, a morphism of PROPs is a pair of functions: one from the colors of $Q$ to the colors of $T$, $\phi_c\colon Col(Q)\to Col(T)$, and for each pair of finite list of colors $\vec{x}=\{x_1,\ldots,x_n\}$ and $\vec{y}=\{y_1,\ldots,y_m\}$ in $Col(Q)$, a function $\phi_m\colon Q(\vec{x},\vec{y})\to T(\phi_c(\vec{x}),\phi_c(\vec{y}))$. The two functions $\phi_c$ and $\phi_m$ 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.

###### Definition

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

$F : T \to C$

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

## Model Structure on Simplicial PROPs

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:

###### Proposition

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 $\phi:Q\to T$ is a weak equivalence (resp. fibration) if for each simplicial set of morphisms the induced map $Q(\vec{x},\vec{y})\to T(\vec{x},\vec{y})$ is also a weak equivalence (resp. fibration).

Recall now that simplicial PROPs admit an obvious forgetful functor to categories that factors through simplicial categories. Denote this functor by $\pi_0\colon sPROP\to Cat$. Using this notation, we have (again from HR2):

###### Proposition

There is a cofibrantly generated model structure on the category of all simplicial PROPs where a morphism of PROPs $\phi\colon Q\to T$ is a weak equivalence (resp. fibration) if:

• the induced morphism of mapping complexes $\phi_m\colon Q(\vec{x},\vec{y})\to T(\phi_c(\vec{x}),\phi_c(\vec{y}))$ is a weak equivalence (resp. Kan fibration) of simplicial sets, and

• the functor $\pi_0\colon\pi_0Q\to \pi_0T$ is a weak equivalence (resp. isofibration) of categories.

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.

## Examples

### Bialgebras

A perhaps paradigmatic example is that there is a $Vect$-enriched prop whose algebras are bialgebras. It should be observed here that there is no $Vect$-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.

### Endomorphism PROP

Given a set of colors $\mathfrak{C}$ and a closed symmetric monoidal category $E$ with a chosen collection of objects $\mathbf{X}=\{X_c\}_{c\in\mathfrak{C}}$, there is an $E$-enriched PROP $End_{\mathbf{X}}$ with morphism $E$-mapping spaces $End_{\mathbf{X}}(\{X_i\},\{Y_j\})=E(\otimes_i X_i,\otimes_j Y_j)$.

## Relation to properads and polycategories

Properads have a similar multiple compositional structure to PROPs, but without the tensor product of morphisms, meaning operations cannot be concatenated. Polycategories are also similar, but only allow composition along a single object at once. See Polycategory: Relation to properads for a more detailed explanation.