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.

Definition

A pseudo-tensor category C consists of the following data

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

  $P_I(\{X_i\},Y),$

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

  $\circ :P_I(\{Y_i\},Z)\times \prod _{i\in I}{P}_{{\pi }^{-1}(i)}(\{X_j\},Y_i)\to P_J(\{X_j\},Z),$

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

subject to the associativity and the unitality axioms

• for surjective maps $K\to J$ and $J\to I$, $\phi \circ ({\psi }_i\circ {\chi }_j)=(\phi \circ {\psi }_i)\circ {\chi }_j$,
• ${\mathrm{id}}_Y\circ \phi =\phi \circ {\mathrm{id}}_{X_i}=\phi$.

Let C be a pseudo-tensor category. For given objects $X,Y$, let $\mathrm{Hom}(X,Y)=P_{*}(\{X\},Y)$. Then the class of objects in $C$ together with Hom, $\circ$ 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 $P_I=\mathrm{\varnothing },\mathrm{\#}I>1$.

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 (unital semigroup) and thus a pseudo-tensor category with a single is like a monoid but with various n-ary operators. A finite set $\{X_i{\}}_{i\in I}$ 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

$P_I(\{X_i\},Y)=\mathrm{Hom}({\otimes }_{i\in I}{X}_i,Y),$

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

${\otimes }_{j\in J}{X}_j\stackrel{\sim }{\to }{\otimes }_{i\in I}({\otimes }_{j\in {\pi }^{-1}(i)}{X}_j)$

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

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.

Linear case

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

$P_I(\{X_i\},Y)=\mathrm{Hom}({\otimes }_{i\in I}{X}_i,Y).$

References

• Ch. 1 of A. Beilinson and V. Drinfeld, "Chiral algebras," [1]
• § 1.4. of Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and Deligne's conjecture". Conférence Moshé Flato 1999: Quantization, Deformations, and Symmetries I. pp. 255–307. ISBN 9780792365402. 

Further reading

• https://mathoverflow.net/questions/134687/pseutotensor-category