Draft:Associativity isomorphism

Associativity isomorphism in category theory

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In mathematics, specifically in the field of category theory, the associativity isomorphism implements the notion of associativity with respect to monoidal products in semi-groupal (or monoidal-without-unit) categories.

Definition

A category, ${\displaystyle {\mathcal {C}}}$ , is called semi-groupal if it comes equipped with a functor ${\displaystyle {\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {C}}}$  such that the pair ${\displaystyle (A,B)\mapsto A\otimes B}$  for ${\displaystyle A,B\in {\text{ob}}({\mathcal {C}})}$ , as well as a collection of natural isomorphisms known as the associativity isomorphisms (or "associators")^[1][2]. These isomorphisms, ${\displaystyle a_{X,Y,Z}:X\otimes (Y\otimes Z)\to (X\otimes Y)\otimes Z}$ , are such that the following "pentagon identity" diagram commutes.

 

Commutative diagram for associativity isomorphism

Applications

In tensor categories

A tensor category^[3], or monoidal category, is a semi-groupal category with an identity object, ${\displaystyle I}$ , such that ${\displaystyle I\otimes A\cong A}$  and ${\displaystyle A\otimes I\cong A}$ . modular tensor categories have many applications in physics, especially in the field of topological quantum field theories^[4][5].

References

1. MacLane, Saunders (1963). "Natural Associativity and Commutativity". Rice Univ. Studies. 49 (4): 28–46.
2. MacLane, Saunders. Categories for the Working Mathematician (2 ed.). p. 162.
3. Barr, Michael; Wells, Charles. Category Theory for Computing Science. p. 419.
4. "Modular tensor category".
5. Rowell, E., Stong, R. & Wang, Z. On Classification of Modular Tensor Categories. Commun. Math. Phys. 292, 343–389 (2009). https://doi.org/10.1007/s00220-009-0908-z