nLab PROP

This article is about the symmetric monoidal category. For the type of h-propositions see Prop.

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Idea

A PROP is a symmetric monoidal category generated by a single object. The term “PROP” is meant to be an abbreviation for “PROducts and Permutations”.

PROPs are 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 non-cartesian contexts. In this respect they are similar to operads. However, PROPs are more general than operads, 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

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

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.

Related concepts

• PRO
• polycategory
• properad
• dioperad

References

• Teimuraz Pirashvili, On the PROP corresponding to bialgebras, Cahiers de Topologie et Géométrie Différentielle Catégoriques 43 (2002) [publication, arXiv:math/0110014]

• Wee Liang Gan, Koszul Duality for Dioperads, Mathematical Research Letters 10, 2003. (doi:10.4310/MRL.2003.v10.n1.a11, arXiv:math/0201074)

• Steve Lack, Composing PROPs, Theory and Applications of Categories 13, 2004. (tac:13-9)

• Donald Yau, Higher dimensional algebras via colored PROPs, 2008. (arXiv:0809.2161).

• Martin Markl, Sergei Merkulov, Sergey Shadrin, Wheeled PROPs, graph complexes and the master equation, Journal of Pure and Applied Algebra 213 (2009) [doi:10.1016/j.jpaa.2008.08.007, arXiv:math/0610683]

• Philip Hackney and Marcy Robertson, On the Category of PROPs, Applied Categorical Structures 23, 2015. (doi:10.1007/s10485-014-9369-4, arXiv:1207.2773)

• Philip Hackney and Marcy Robertson, The Homotopy Theory of Simplicial PROPs, Israel Journal of Mathematics 219, 2017. (doi:10.1007/s11856-017-1500-4, arXiv:1209.1087)

• John Baez, Brandon Coya and Franciscus Rebro, Props in network theory, Theory and Applications of Categories 33, 2018. (tac:33-25)

On classifying spaces for algebras over a PROP:

• Sinan Yalin, Classifying spaces and moduli spaces of algebras over a prop, Algebr. Geom. Topol. 14 (2014) 2561-2593 [arXiv:1207.2964v1, doi:10.2140/agt.2014.14.2561]