nLab balanced monoidal category

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Balanced monoidal categories

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Balanced monoidal categories

Definition

A twist, or balance, in a braided monoidal category 𝒞\mathscr{C} is a natural isomorphism θ\theta from the identity functor on 𝒞\mathscr{C} to itself satisfying the following compatibility condition with the braiding β\beta:

θ AB=β B,Aβ A,B(θ Aθ B),A,B𝒞 \theta_{A\otimes B} \;=\; \beta_{B,A} \circ \beta_{A,B} \circ (\theta_A \otimes \theta_B), \,\, \forall A, B \in \mathscr{C}

A balanced monoidal category is a braided monoidal category equipped with such a balance.

Beware that there is an un-related notion of balanced categories.

Properties

In the language of string diagrams, the balancing is represented by a 360-degree twist:

Every symmetric monoidal category is balanced in a canonical way. In fact, the identity natural transformation on the identity functor of 𝒞\mathscr{C} is a balance on 𝒞\mathscr{C}. In this way, the twist can be seen as a way of “controlling” the non-symmetric behavior of the braiding.

A braided rigid monoidal category is balanced if and only if it is a pivotal category, but a balanced monoidal category need not be rigid (cf. Selinger 2011, Lem. 4.20).

References

The original definition:

The above definition follows:

  • Jeff Egger, Appendix C in: Of Operator Algebras and Operator Spaces (2006) [pdf]

See also:

Last revised on February 21, 2024 at 05:25:13. See the history of this page for a list of all contributions to it.