Pseudo-tensor category

In mathematics, specifically category theory, a pseudo-tensor category is a generalization of a symmetric monoidal category (also known as a tensor category) introduced by A. Beilinson and V. Drinfeld in their book "Chiral algebras".

The notion can also be defined as a colored operad or multicategory. In particular, a pseudo-tensor category with a single object is the same as an operad.

A pseudo-tensor category C consists of the following data¹

• A class of objects,
• For each finite set ${\displaystyle I}$, each finite set of objects ${\displaystyle X_{i},i\in I}$ parametrized by ${\displaystyle I}$ and another object ${\displaystyle Y}$, the set

  ${\displaystyle P_{I}(\{X_{i}\},Y),}$

• For each surjective map ${\displaystyle J\to I}$ between finite sets, finite sets of objects ${\displaystyle \{Y_{i}\}_{i\in I},\{X_{j}\}_{j\in J}}$ and an object Z, the map

  ${\displaystyle \circ :P_{I}(\{Y_{i}\},Z)\times \prod _{i\in I}P_{\pi ^{-1}(i)}(\{X_{j}\},Y_{i})\to P_{J}(\{X_{j}\},Z),}$

• For each object ${\displaystyle X}$, the element ${\displaystyle \operatorname {id} _{X}}$ in ${\displaystyle P_{*}(\{X\},X)}$ where * is a set with a single element,

subject to the associativity and the unitality axioms

• for surjective maps ${\displaystyle K\to J}$ and ${\displaystyle J\to I}$, ${\displaystyle \varphi \circ (\psi _{i}\circ \chi _{j})=(\varphi \circ \psi _{i})\circ \chi _{j}}$,
• ${\displaystyle \operatorname {id} _{Y}\circ \varphi =\varphi \circ \operatorname {id} _{X_{i}}=\varphi }$.

Let C be a pseudo-tensor category. For given objects ${\displaystyle X,Y}$, let ${\displaystyle \operatorname {Hom} (X,Y)=P_{*}(\{X\},Y)}$. Then the class of objects in ${\displaystyle C}$ together with Hom, ${\displaystyle \circ }$ and the identities form a category. Thus, a pseudo-tensor category can be thought of as a category together with extra data. In particular, a category is the same thing as a pseudo-tensor category with ${\displaystyle P_{I}=\emptyset ,\#I>1}$.²

On the other extreme, a pseudo-tensor category with a single object is the same as an operad.³ Indeed, a category with a single object is a monoid (unital semigroup) and thus a pseudo-tensor category with a single is like a monoid but with various n-ary operators. A finite set ${\displaystyle \{X_{i}\}_{i\in I}}$ in the definition of a pseudo-tensor is an unordered finite set. This amounts to the invariance under a symmetric group in the definition of an operad.

Finally, let C be a symmetric monoidal category. Then let

${\displaystyle P_{I}(\{X_{i}\},Y)=\operatorname {Hom} (\otimes _{i\in I}X_{i},Y),}$

which is well-defined since C is symmetric. The symmetric-monoidal structure include coherent isomorphisms

${\displaystyle \otimes _{j\in J}X_{j}{\overset {\sim }{\to }}\otimes _{i\in I}(\otimes _{j\in \pi ^{-1}(i)}X_{j})}$

which gives ${\displaystyle \circ }$ in the definition of a pseudo-tensor category. Conversely, a pseudo-tensor category with such ${\displaystyle \otimes _{i\in I}X_{i}}$ and coherent isomorphisms defines a symmetric monoidal category. In this way, a pseudo-tensor category generalizes a symmetric monoidal category.⁴

In the definition, we can drop the symmetry requirement; namely, instead of a finite set of objects, we can use a finite sequence of objects. In this case, we get the notion of a multicategory. In other words, a pseudo-tensor category is (essentially) a symmetric multicategory.

Like an enriched category, a pseudo-tensor category can also be defined over a symmetric monoidal category V; namely, we require ${\displaystyle P_{I}}$ as well as ${\displaystyle \circ }$ take values in V instead of the category of sets in the definition. A particularly important case is when V is the category of vector spaces; i.e., the images of ${\displaystyle P_{I}}$ are sets of multilinear maps and if tensor product is available,

${\displaystyle P_{I}(\{X_{i}\},Y)=\operatorname {Hom} (\otimes _{i\in I}X_{i},Y).}$