Rigid category

Uses

One important application of rigidity is in the definition of the trace of an endomorphism of a rigid object. The trace can be defined for any pivotal category, i. e. a rigid category such that ( )^**, the functor of taking the dual twice repeated, is isomorphic to the identity functor. Then for any right rigid object X, and any other object Y, we may define the isomorphism

${\displaystyle {\varphi }_{X,Y}:\{\begin{array}{rcl}\mathrm{H}\mathrm{o}\mathrm{m}(\mathbf{1},X^{\ast }\otimes Y)& ⟶& \mathrm{H}\mathrm{o}\mathrm{m}(X,Y)\\ f& ⟼& ({ϵ}_X\otimes i{d}_Y)\circ (i{d}_X\otimes f)\end{array}}$

and its reciprocal isomorphism

${\displaystyle {\psi }_{X,Y}:\{\begin{array}{rcl}\mathrm{H}\mathrm{o}\mathrm{m}(X,Y)& ⟶& \mathrm{H}\mathrm{o}\mathrm{m}(\mathbf{1},X^{\ast }\otimes Y)\\ g& ⟼& (i{d}_{X^{\ast }}\otimes g)\circ {\eta }_X\end{array}}$.

Then for any endomorphism ${\displaystyle f:X\to X}$, the trace is of f is defined as the composition:

${\displaystyle \mathrm{t}\mathrm{r}f:\mathbf{1}\stackrel{{\psi }_{X,X}(f)}{\to }{X}^{\ast }\otimes X\stackrel{{\gamma }_{X,X}}{\to }X\otimes X^{\ast }\stackrel{{ϵ}_X}{\to }\mathbf{1},}$

We may continue further and define the dimension of a rigid object to be:

${\displaystyle \dim X:=\mathrm{t}\mathrm{r}{\mathrm{i}\mathrm{d}}_X}$.

Rigidity is also important because of its relation to internal Hom's. If X is a left rigid object, then every internal Hom of the form [X, Z] exists and is isomorphic to Z ⊗ Y. In particular, in a rigid category, all internal Hom's exist.

Alternative terminology

A monoidal category where every object has a left (respectively right) dual is also sometimes called a left (respectively right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.

Discussion

A monoidal category is a category with a tensor product, precisely the sort of category for which rigidity makes sense.

The category of pure motives is formed by rigidifying the category of effective pure motives.