In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X^* (the internal Hom [X, 1]) and a morphism 1 → X ⊗ X^* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined (following Alexander Grothendieck) by Neantro Saavedra Rivano in his thesis on Tannakian categories.

There are at least two equivalent definitions of a rigidity.

${\displaystyle X~{\xrightarrow {\eta _{X}\otimes \mathrm {id} _{X}}}~(X\otimes Y)\otimes X~{\xrightarrow {\alpha _{X,Y,X}^{-1}}}~X\otimes (Y\otimes X)~{\xrightarrow {\mathrm {id} _{X}\otimes \epsilon _{X}}}~X}$

${\displaystyle Y~{\xrightarrow {\mathrm {id} _{Y}\otimes \eta _{X}}}~Y\otimes (X\otimes Y)~{\xrightarrow {~\alpha _{X,Y,X}~}}~(Y\otimes X)\otimes Y~{\xrightarrow {\epsilon _{X}\otimes \mathrm {id} _{Y}}}~Y}$

are identities. A right rigid object is defined similarly.

An inverse is an object X⁻¹ such that both X ⊗ X⁻¹ and X⁻¹ ⊗ X are isomorphic to 1, the identity object of the monoidal category. If an object X has a left (respectively right) inverse X⁻¹ with respect to the tensor product then it is left (respectively right) rigid, and X^* = X⁻¹.

The operation of taking duals gives a contravariant functor on a rigid category.

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