Pseudo-tensor category


In mathematics, specifically category theory, a pseudo-tensor category is a generalization of a symmetric monoidal 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.

Definition

A pseudo-tensor category ''C consists of the following data
  • A class of objects,
  • For each finite set, each finite set of objects parametrized by and another object, the set
  • :
  • For each surjective map between finite sets, finite sets of objects and an object Z'', the map
  • :
  • For each object, the element in where * is a set with a single element,
subject to the associativity and the unitality axioms
  • for surjective maps and,,
  • .
Let C be a pseudo-tensor category. For given objects, let. Then the class of objects in together with Hom, 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.
On the other extreme, a pseudo-tensor category with a single object is the same as an operad. Indeed, a category with a single object is a monoid and thus a pseudo-tensor category with a single is like a monoid but with various n-ary operators. A finite set 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
which is well-defined since C is symmetric. The symmetric-monoidal structure include coherent isomorphisms
which gives in the definition of a pseudo-tensor category. Conversely, a pseudo-tensor category with such and coherent isomorphisms defines a symmetric monoidal category. In this way, a pseudo-tensor category generalizes a symmetric monoidal category.
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 a symmetric multicategory.

Linear case

Like an enriched category, a pseudo-tensor category can also be defined over a symmetric monoidal category V; namely, we require as well as 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 are sets of multilinear maps and if tensor product is available,