nLab balanced monoidal category

Balanced monoidal categories


Monoidal categories

monoidal categories

With symmetry

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 BB is a natural transformation from the identity functor on BB to itself satisfying a certain condition that links it to the braiding. A balanced monoidal category is a braided monoidal category equipped with such a balance.

The condition linking the balancing to the braiding, where θ\theta is the balance and β\beta is the braiding, is that θ xy\theta_{x \otimes y} should be the composite of β x,y\beta_{x,y}, θ yθ x\theta_y \otimes \theta_x, and β y,x\beta_{y,x}.

A balanced monoidal category is a special case of a balanced pseudomonoid in a balanced monoidal bicategory?.


Every symmetric monoidal category is balanced in a canonical way; in fact, the identity natural transformation (on the identity functor of BB) is a balance on BB if and only if BB is symmetric. Thus balanced monoidal categories fall between braided monoidal categories and symmetric monoidal categories. (They should not be confused with balanced categories, which are unrelated.)

In the string diagram calculus for ribbon categories, the twist is represented by a 360-degree twist in a ribbon.


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 January 17, 2020 at 23:09:14. See the history of this page for a list of all contributions to it.