In mathematics, the skeletonization of fusion categories is a process whereby one extracts the core data of a fusion category or related categorical object in terms of minimal set-theoretic information. This set-theoretic information is referred to as the skeletal data of the fusion category. This process is related to the general technique of skeletonization in category theory. Skeletonization is often used for working with examples, doing computations, and classifying fusion categories. The relevant feature of fusion categories which makes the technique of skeletonization effective is the strong finiteness conditions placed on fusion categories, such as the requirements that they have finitely many isomorphism classes of simple objects and that all of their hom-spaces are finite dimensional. This allows the entire categorical structure of a fusion category to be encoded in a finite amount of complex numbers, arranged into tensors. The coherence conditions on fusion categories turn into compatibility conditions on the tensors. In this context, skeletonization is the opposite process of categorification, which takes set-theoretic information and turns it into category-theoretic data.
For fusion categories The skeletonization of fusion categories is often stated in terms of string diagrams. In this approach, morphims in the category are depicted as strings, which one can interpret as spacetime trajectories of some point-like objects.
The tensor product is denoted by placing strings adjacent to one another.
Let C {\displaystyle {\mathcal {C}}} denote a fusion category. Let L {\displaystyle {\mathcal {L}}} denote the set of isomorphism classes of simple objects of C {\displaystyle {\mathcal {C}}} . By the definition of a fusion category, L {\displaystyle {\mathcal {L}}} is a finite set and contains a distinguished element [ 1 ] {\displaystyle [{\bf {1}}]} corresponding to the tensor unit. Since fusion categories are semi-simple, for all [ A ] , [ B ] ∈ L {\displaystyle [A],[B]\in {\mathcal {L}}} , there is a decomposition A ⊗ B ≅ ⨁ [ C ] ∈ L N C A , B ⋅ C {\textstyle A\otimes B\cong \bigoplus _{[C]\in {\mathcal {L}}}N_{C}^{A,B}\cdot C} . Here, the coefficient N C A , B {\displaystyle N_{C}^{A,B}} describes with which multiplicity C {\displaystyle C} occurs in the tensor product of A {\displaystyle A} and B {\displaystyle B} . These coefficients N C A , B {\textstyle N_{C}^{A,B}} are non-negative integers which only depend on the isomorphism classes of A , B , C ∈ C {\displaystyle A,B,C\in {\mathcal {C}}} , and are referred to as the fusion coefficients of C {\displaystyle {\mathcal {C}}} , and are the first basic piece of the skeletal data of C {\displaystyle {\mathcal {C}}} . Given simple objects A , B , C ∈ C {\displaystyle A,B,C\in {\mathcal {C}}} , any morphisms η : C → A ⊗ B {\displaystyle \eta :C\to A\otimes B} can be depicted using string diagrams notion as follows.
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