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 {\displaystyle I} , each finite set of objects X i , i ∈ I {\displaystyle X_{i},i\in I} parametrized by I {\displaystyle I} and another object Y {\displaystyle Y} , the set
P I ( { X i } , Y ) , {\displaystyle P_{I}(\{X_{i}\},Y),}
For each surjective map J → I {\displaystyle J\to I} between finite sets, finite sets of objects { Y i } i ∈ I , { X j } j ∈ J {\displaystyle \{Y_{i}\}_{i\in I},\{X_{j}\}_{j\in J}} and an object Z, the map
∘ : P I ( { Y i } , Z ) × ∏ i ∈ I P π − 1 ( i ) ( { X j } , Y i ) → P J ( { X j } , Z ) , {\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 X {\displaystyle X} , the element id X {\displaystyle \operatorname {id} _{X}} in P ∗ ( { X } , X ) {\displaystyle P_{*}(\{X\},X)} where * is a set with a single element, subject to the associativity and the unitality axioms
for surjective maps K → J {\displaystyle K\to J} and J → I {\displaystyle J\to I} , φ ∘ ( ψ i ∘ χ j ) = ( φ ∘ ψ i ) ∘ χ j {\displaystyle \varphi \circ (\psi _{i}\circ \chi _{j})=(\varphi \circ \psi _{i})\circ \chi _{j}} ,
id Y ∘ φ = φ ∘ id X i = φ {\displaystyle \operatorname {id} _{Y}\circ \varphi =\varphi \circ \operatorname {id} _{X_{i}}=\varphi } . Let C be a pseudo-tensor category. For given objects X , Y {\displaystyle X,Y} , let Hom ( X , Y ) = P ∗ ( { X } , Y ) {\displaystyle \operatorname {Hom} (X,Y)=P_{*}(\{X\},Y)} . Then the class of objects in C {\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 P I = ∅ , # I > 1 {\displaystyle P_{I}=\emptyset ,\#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 ∈ I {\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
P I ( { X i } , Y ) = Hom ( ⊗ i ∈ I X i , Y ) , {\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
⊗ j ∈ J X j → ∼ ⊗ i ∈ I ( ⊗ j ∈ π − 1 ( i ) X j ) {\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 ⊗ i ∈ I X i {\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. 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 {\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 P I {\displaystyle P_{I}} are sets of multilinear maps and if tensor product is available,
P I ( { X i } , Y ) = Hom ( ⊗ i ∈ I X i , Y ) . {\displaystyle P_{I}(\{X_{i}\},Y)=\operatorname {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. arXiv:math/0001151. ISBN 9780792365402.
Further reading https://mathoverflow.net/questions/134687/pseutotensor-category