nLab rigid monoidal category



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



A rigid (monoidal) category, also called an autonomous (monoidal) category is a kind of category with duals. In particular, a rigid monoidal category is a category in which every object AA has a dual object A *A^*. Depending on whether these duals exist on the right or the left we call the category left or right rigid. Sometimes rigid refers to categories with duals on both sides.

It is important to note that every right/left rigid category can be fully embedded into a category with duals on both sides, by repeatedly applying the double dual functor and taking a colimit (see e.g. Selinger, Proposition 4.7).


A monoidal category is right rigid (or right autonomous) if every object has a right dual. Similarly, a monoidal category is left rigid (or left autonomous) if every object has a left dual. A monoidal category is autonomous (sometimes called rigid) if duals exist on both sides.

Conventions differ regarding which type of duals are which. One convention is as follows: a right dual of an object AA in a monoidal category π’ž\mathscr{C} is an object A *A^* equipped with evaluation/co-evaluation maps

ev A:AβŠ—A *β†’1,coev A:1β†’A *βŠ—A\text{ev}_{A}:A\otimes A^*\to 1,\,\, \text{coev}_{A}:1\to A^*\otimes A

where 11 is the tensor unit, subject the the triangle identities (snake diagrams)

(ev AβŠ—id A)∘(id AβŠ—coev A)=id A,(\text{ev}_A\otimes \text{id}_A)\circ (\text{id}_A\otimes \text{coev}_{A})=\text{id}_{A},
(id A *βŠ—ev A)∘(coev AβŠ—id A *)=id A *.(\text{id}_{A^*}\otimes \text{ev}_{A})\circ ( \text{coev}_{A}\otimes \text{id}_{A^*})=\text{id}_{A^*}.

A left dual is an object A *A^* paired with morphisms ev A:A *βŠ—Aβ†’1\text{ev}_A:A^*\otimes A\to 1 and coev A:1β†’AβŠ—A *\text{coev}_A:1\to A\otimes A^* satisfying the dual conditions.


The canonical example of a rigid category is the category Vec k\text{Vec}_k of finite dimensional vector spaces over a field kk. Namely, given a vector space VV the standard dual V *=Hom k(V,k)V^*=\text{Hom}_k(V,k) serves as a categorical dual for VV. Explicitely we have the following:


Let kk be a field. For every vector space V∈Vec kV\in \text{Vec}_k, define the maps

ev V:VβŠ—V *β†’k\text{ev}_V: V\otimes V^*\to k
(v,Ο†)↦φ(v)(v,\varphi)\mapsto \varphi(v)


coev V:kβ†’V *βŠ—V\text{coev}_V: k\to V^*\otimes V
x↦xβ‹…βˆ‘ i∈IΟ† iβŠ—v ix\mapsto x\cdot \sum_{i\in I}\varphi_i\otimes v_i

where {v i} i∈I\{v_i\}_{i\in I} is a basis for VV, and Ο† i\varphi_i is a dual basis. That is, Ο† i\varphi_i is defined by

Ο† i(v j)={1 i=j 0 otherwise.\varphi_i(v_j)= \begin{cases} 1 & i=j\\ 0 & \text{otherwise}. \end{cases}

These formulas give well defined Vec k\text{Vec}_k-morphisms. Moreover, the satisfy the snake identities and hence give Vec k\text{Vec}_k the structure of a (right) rigid category.


To begin, we verify that the definition of coev\text{coev} is independent of basis. Suppose {v˜ i} i∈I\{\tilde{v}_i\}_{i\in I} is a different choice of basis. There is a change of basis matrix c=(c i,j) (i,j)∈IΓ—Ic=(c_{i,j})_{(i,j)\in I\times I} between them. It is straightforward to see that the dual bases are related by the change of basis matrix c βˆ’1c^{-1}. Expanding, we thus find that

βˆ‘ iv˜ iβŠ—Ο†Λœ i=βˆ‘ i(βˆ‘ jc i,jβ‹…v i)βŠ—(βˆ‘ k(c βˆ’1) k,iβ‹…Ο† i)\sum_{i}\tilde{v}_i\otimes \tilde{\varphi}_i=\sum_{i}\left(\sum_{j}c_{i,j}\cdot v_i\right)\otimes \left(\sum_{k}(c^{-1})_{k,i}\cdot \varphi_i\right)
=βˆ‘ j,k(βˆ‘ ic i,j(c βˆ’1) k,i)β‹…v iβŠ—Ο† i.=\sum_{j,k} \left(\sum_{i}c_{i,j}(c^{-1})_{k,i}\right)\cdot v_i\otimes \varphi_i.

The definition of the inverse matrix says that βˆ‘ ic i,j(c βˆ’1) k,i={1 j=k 0 otherwise\sum_{i}c_{i,j}(c^{-1})_{k,i}=\begin{cases}1 & j=k \\ 0 &\text{otherwise}\end{cases}. Hence, we conclude that

βˆ‘ iv˜ iβŠ—Ο†Λœ i=βˆ‘ iv iβŠ—Ο† i,\sum_{i}\tilde{v}_i\otimes \tilde{\varphi}_i=\sum_{i}v_i\otimes \varphi_i,

so coev V\coev_{V} is well defined as desired. We now show that the rigidity diagrams commute. Going around the square, we find that this is equivalent to the condition that 1βŠ—w=βˆ‘ iΟ† i(w)βŠ—v i1\otimes w=\sum_{i}\varphi_i(w)\otimes v_i for all w∈Vw\in V. This follows from choosing a basis such that v i=wv_i=w for some ii. Then, Ο† i(w)βŠ—v i=w\varphi_i(w)\otimes v_i=w and all the other terms are zero, making the equality obvious.

Another key example is that of endofunctor categories. Given any category π’ž\mathscr{C}, the hom-space of functors Fun(π’ž,π’ž)\text{Fun}(\mathscr{C},\mathscr{C}) is a monoidal category, with the monoidal structure coming from composition. Duals correspond to adjoint functor as can be stated formally below:


Let π’ž\mathscr{C} be a category, and let F∈Fun(π’ž,π’ž)F\in \text{Fun}(\mathscr{C},\mathscr{C}) be an endofunctor. Let F *F^* be the right adjoint of FF. F *F^* is a right dual for FF, under the natural transformations

ev F:FβŠ—F *β†’id π’ž\text{ev}_{F}:F\otimes F^*\to \text{id}_{\mathscr{C}}
F(F *(A))β†’id F *(A)A,Aβˆˆπ’žF(F^*(A))\xrightarrow{\text{id}_{F^*(A)}} A,\,\, A\in \mathscr{C}

where we here make the implicit identification

Hom π’ž(F(F *(A)),A)β‰…Hom π’ž(F *(A),F *(A))\text{Hom}_{\mathscr{C}}(F(F^*(A)),A)\cong\text{Hom}_{\mathscr{C}}(F^*(A),F^*(A))

using adjointness, and

coev F:id π’žβ†’FβŠ—F *\text{coev}_{F}:\text{id}_{\mathscr{C}}\to F\otimes F^*
Aβ†’id F(A)F *(F(A)),Aβˆˆπ’žA\xrightarrow{\text{id}_{F(A)}} F^*(F(A)),\,\, A\in \mathscr{C}

where we here make the implicit identification

Hom π’ž(A,F *(F(A)))β‰…Hom π’ž(F(A),F(A))\text{Hom}_{\mathscr{C}}(A,F^*(F(A)))\cong\text{Hom}_{\mathscr{C}}(F(A),F(A))

using adjointness. Conversely, if F *F^* is a right dual for an endofunctor FF then F *F^* is a right adjoint for FF.


Verifying the details is straightforward.

Graphical language

Rigid categories are best expressed using the language of string diagrams. That is, we can write the rigidity axioms as

Even better, we often fix graphical notation for ev\text{ev} and coev\text{coev} which makes these axioms even simpler to write out:

We can now express rigidity as follows:

Rigidity is thus graphically the statement that string diagrams can be β€œstraightened out”. This approach is taken, for example, in Selinger.


Note that this definition only asserts the existence of the dual objects. It does not assert that specific duals have been chosen. However, the choice of duals is unique up to unique isomorphism, justifying reference to β€˜the’ dual of an object; in fact, this extends to a contravariant anafunctor *:Mβ†’M{}^*\colon M \to M. (Using the axiom of choice to pick duals for every object at once, we can make this into a strict functor.) This functor is defined explicitly by the formula

f *:A *β†’coev BβŠ—id A *B *βŠ—BβŠ—A *β†’id B *βŠ—fβŠ—id A *B *βŠ—AβŠ—A *β†’id B *βŠ—ev AB *f^*:A^*\xrightarrow{\text{coev}_B\otimes \text{id}_{A^*}} B^*\otimes B\otimes A^*\xrightarrow{\text{id}_{B^*}\otimes f\otimes \id_{A^*}} B^*\otimes A\otimes A^* \xrightarrow{\text{id}_{B^*}\otimes \text{ev}_A} B^*

This formula is clearer when written graphically:

Additionally, this definition does not assert that the right dual of an object is isomorphic to its left dual: this need not be the case in general, though it is true in a braided monoidal category, and thus automatically also in a symmetric monoidal category (this last fact can be considered an algebraic form of the β€œWhitney trick” for knots; see this MO discussion). Left duals and right duals are also isomorphic in a semisimple category. Note that a rigid monoidal category which is also symmetric is sometimes called compact closed, or simply β€œcompact”.

In practice, algebraic geometers are the most frequent users of the term β€˜rigid’, and they focus on the symmetric monoidal case, so they ignore the difference between right and left duals.


Tannaka duality

The statement of Tannaka duality for associative algebras says that rigid monoidal categories equipped with a fiber functor are categories of modules over a Hopf algebra.

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

Free rigid monoidal categories

The inclusion of the 2-category of monoidal categories into the 2-category of rigid monoidal categories admits a left 2-adjoint functor? LL.

Furthermore, the unit of the adjunction is a strong monoidal fully faithful functor, i.e., any monoidal category CC admits a fully faithful strong monoidal functor C→L(C)C\to L(C), where L(C)L(C) is a rigid monoidal category.

See Theorems 1 and 2 in Delpeuch.


  • N. Saavedra Rivano, β€œCatΓ©gories Tannakiennes.” Bulletin de la SociΓ©tΓ© MathΓ©matique de France 100 (1972): 417-430. EuDML

  • Antonin Delpeuch, Autonomization of monoidal categories, arXiv, doi.

  • Peter Selinger, A survey of graphical languages for monoidal categories, New structures for physics, arXiv

Last revised on April 8, 2024 at 15:33:40. See the history of this page for a list of all contributions to it.