nLab
category with duals

Categories with duals

Context

Monoidal categories

monoidal categories

With symmetry

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

Categories with duals

Idea

A category with duals is a category where objects and/or morphisms have duals. This exists in several flavours; this list is mostly taken from a recent categories list post from Peter Selinger.

Categories with duals for objects

Categories with duals for morphisms

One might write something about these too, or put them on a separate page. In the meantime, see the table of contents to the right.

There at least two commonspread kinds of categories with duals for morphisms:

  • dagger categories where each morphism f:XYf:X \to Y has a \dagger-dual f :YXf^\dagger : Y \to X, without any extra property.
  • groupoids, where each morphism f:XYf:X \to Y has an inverse f 1:YXf^{-1} :Y \to X defined by the properties ff 1=1 Yf f^{-1} = 1_Y, f 1f=1 Xf^{-1}f = 1_X.

Moreover, every category enriched in one of the kind of categories listed above will have a notion of ‘dual’ for its morphisms.

References

Last revised on October 21, 2021 at 05:19:05. See the history of this page for a list of all contributions to it.