nLab balanced monoidal category

Balanced monoidal categories


Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Balanced monoidal categories


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

ΞΈ AβŠ—B=Ξ² B,A∘β A,B∘(ΞΈ AβŠ—ΞΈ B),βˆ€A,Bβˆˆπ’ž\theta_{A\otimes B}=\beta_{B,A}\circ \beta_{A,B}\circ (\theta_A\otimes \theta_B),\,\, \forall A,B\in\mathscr{C}

where Ξ²\beta is the braiding on π’ž\mathscr{C}. A balanced monoidal category is a braided monoidal category equipped with such a balance. Equivalently, a balanced monoidal category can be described as a braided pivotal category.

Balanced monoidal categories should not be confused with the other unrelated notation of a balanced category.


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

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


This definition is taken from Jeff Egger (Appendix C), but the original definition can be found in chapter 4 of this paper by Joyal and Street:

  • A. Joyal, R. Street, The geometry of tensor calculus I, Adv. Math. 88(1991), no. 1, 55–112, doi.

Last revised on May 8, 2023 at 00:10:19. See the history of this page for a list of all contributions to it.