Associativity isomorphism

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] [ full citation needed ]} 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] [ full citation needed ]} 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,^{[ speculation? ]} especially in the field of topological quantum field theories. ^{[4] [ unreliable source? ] [5] [ dubious – discuss ]}

References

1. MacLane, Saunders (1963). "Natural Associativity and Commutativity". Rice Univ. Studies. 49 (4): 28–46. hdl:1911/62865.
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, Eric; Stong, Richard; Wang, Zhenghan (2009). "On Classification of Modular Tensor Categories". Communications in Mathematical Physics. 292 (2): 343–389. Bibcode:2009CMaPh.292..343R. doi: 10.1007/s00220-009-0908-z .