duoidal category


Monoidal categories

Duoidal categories


A duoidal category is a category with two monoidal structures which interchange laxly.


Duoidal categories

A duoidal category, or 2-monoidal category, is a pseudomonoid in the 2-category MonCat lMonCat_l of monoidal categories and lax monoidal functors. Thus it is a monoidal category, say (C,,I)(C,\diamond,I), together with an additional monoidal structure (C,,J)(C,\star,J) such that :C×CC\star:C\times C\to C and J:1CJ:1\to C are lax monoidal functors with respect to (,I)(\diamond,I) and the coherence axioms of (C,,I)(C,\diamond,I) are monoidal natural transformations with respect to (,I)(\diamond,I). The laxity of \star consists of natural transformations

(AB)(CD)(AC)(BD) (A\star B) \diamond (C\star D) \to (A\diamond C) \star (B\diamond D)


III I\to I\star I

while the laxity of JJ consists of transformations

JJJ J\diamond J \to J


IJ. I \to J.

There are then various axioms, which in particular say that JJ is a \diamond-monoid and II is a \star-comonoid.

It is equivalent to ask that (C,,I)(C,\diamond,I) is a pseudomonoid structure on (C,,J)(C,\star,J) in the 2-category MonCat cMonCat_c of monoidal categories and colax monoidal functors.

The map IJI\to J is actually determined by the rest of the structure; it is the composite

I(JI)(IJ)(JI)(IJ)JI \xrightarrow{\cong} (J\star I)\diamond (I\star J) \to (J\diamond I) \star (I\diamond J) \xrightarrow{\cong} J

as well as the dual composite. If this map is an isomorphism, the duoidal category is called normal.

Duoidal functors

There are three natural kinds of functors between duoidal categories, which are compatibly lax or colax with respect to \star and \diamond. The possible choices of lax and colax seem like there should be four, but in fact only three of them work; in the fourth case the compatibility condition would involve wrong-way composites due to the non-invertibility of the duoidal interchange constraints.

Virtual duoidal categories

A virtual duoidal category is a pseudomonoid in the 2-category of multicategories. That is, it is a multicategory with a monoidal structure (C,,J)(C,\star,J): the \diamond-monoidal structure is replaced by the multicategory structure. Since every monoidal category can be regarded as a representable multicategory, and multicategory functors correspond exactly to lax monoidal functors, duoidal categories can be identified with virtual duoidal categories in which the multicategory structure is representable.

In particular, note that JJ in a virtual duoidal category is a functor of multicategories 1C1\to C, i.e. precisely a monoid in the multicategory CC. As part of this monoid structure it has a unit map ()J()\to J; we say that the virtual duoidal category CC is normal if this map exhibits JJ as a unit object for the multicategory structure.

Nearly all the definitions given below for duoidal categories (except for coduoids and colax/colax functors) make sense just as well for virtual ones.


  • Any braided monoidal category can be made into a (normal) duoidal category in which the two monoidal structures agree. The interchange transformation is built out of the braiding. Indeed, by the Eckmann-Hilton argument, a duoidal category whose above structure transformations are invertible necessarily arises in this way (up to equivalence).

  • Similarly but somewhat more generally, an 2-monoidal category, as a special case of iterated monoidal category (in the sense of Balteanu, Fiedorowicz, Schwänzl, and Vogt), is a duoidal category. More precisely, 2-monoidal categories are the duoidal categories such that \star and JJ are normal lax monoidal functors, meaning concretely that the morphisms IIII\to I\star I and IJI\to J are isomorphisms.

  • If (C,,I)(C,\diamond,I) is a monoidal category with finite products, then it becomes duoidal with \star the cartesian monoidal structure. Dually, if (C,,J)(C,\star,J) is a monoidal category with finite coproducts, then it becomes duoidal with \diamond the cocartesian monoidal structure.

  • If AA and BB are monoidal categories, with AA small and BB cocomplete and closed, then the functor category [A,B][A,B] is duoidal with \star the pointwise tensor product, (XY)(a)=X(a)Y(a)(X\star Y)(a) = X(a) \otimes Y(a), and \diamond the Day convolution tensor product.

  • In particular, if AA is a monoid (in Set) and BB a cocomplete closed monoidal category, then the category B AB^A of “AA-graded objects of BB” is duoidal with \star the pointwise (or “Hadamard”) product and \diamond the “Cauchy” product, (XY) a= b+c=aX aY b(X\diamond Y)_a = \sum_{b+c=a} X_a \otimes Y_b. If BB is pointed, then the interchange law in this case is naturally a split mono, and its canonical retraction gives B AB^A a different duoidal structure with the roles of \star and \diamond switched.

  • Similarly, the category [core(FinSet),Vect][core(FinSet),Vect] of vector species has both a “Hadamard product” (pq)(I)=p(I)q(I)(p\diamond q)(I) = p(I) \otimes q(I). and a “Cauchy product” (pq)(I)= I=STp(S)q(T)(p\star q)(I) = \bigoplus_{I = S\sqcup T} p(S) \otimes q(T), and is duoidal in two ways.

  • The category FF(C)FF(C) of functorial factorizations on a category CC is a duoidal category. The \star product of functorial factorizations FF and FF' is obtained by FF-factoring a morphism and then FF'-factoring the first factor, while the \diamond product is obtained by FF-factoring a morphism and then FF'-factoring the second factor.

  • The category C=[V,V]C=[V,V] of endofunctors of a closed symmetric monoidal category VV is virtually duoidal, with \star being functor composition the multicategory structure being “the one that would be represented by Day convolution if it existed”. That is, a morphism (F,G)H(F,G) \to H is a natural transformation FGHF\otimes G \to H\circ \otimes between functors V×VVV\times V \to V, and so on. This multicategory structure is not generally representable, since the convolution product requires colimits of the size of the domain category, but VV can’t in general be expected to have VV-sized colimits. However, if VV is monoidally locally presentable, then we can restrict to the subcategory End κ(V)End_\kappa(V) of κ\kappa-accessible endofunctors, and in this case there is no problem, since End κ(V)[V κ,V]End_\kappa(V) \simeq [V_\kappa,V] where V κV_\kappa is the essentially small subcategory of κ\kappa-presentable objects. If V=SetV=Set then this (virtual) duoidal structure is normal with I=JI=J being the identity functor, but for general VV it need not be normal.

  • As special case of the preceding when V=SetV=Set and κ=ω\kappa=\omega, consider the functor category [FinSet,Set][FinSet,Set], which is equivalent to the category [Set,Set] f[Set,Set]_f of finitary endofunctors? of SetSet. Since the composite of finitary endofunctors is finitary, we have an induced composition monoidal structure \star. We can define \diamond by convolution with the monoidal structure of FinSetFinSet, making [FinSet,Set][FinSet,Set] duoidal.

  • More generally, instead of End κ(V)=[V κ,V]End_\kappa(V) = [V_\kappa,V] we can consider any category [V,V][V',V] where VV' is a small monoidal subcategory of VV as long as the category of functors VVV\to V that are left Kan extended from VV' is closed under composition. For instance, if AA is a monoidal category and V=[A op,Set]V=[A^{op},Set] with V=AV'=A, then [V,V]Prof(A,A)[V',V] \simeq Prof(A,A) is the category of endo-profunctors of AA, which is therefore duoidal. More generally, the category of endo-proarrows of any monoidal object in a proarrow equipment is duoidal by a similar construction: \star is proarrow composition and \diamond is convolution with the monoidal structure on both sides.

  • We can do something similar for generalized operads with respect to any monoidal pseudomonad on Prof. For instance, with the pseudomonad for symmetric monoidal categories, we obtain a duoidal structure on the category of collections.

Structures in duoidal categories

Bimonoids, bicomonoids, and duoids

In a duoidal category CC, the category Mon (C)Mon_\diamond(C) of monoid objects with respect to \diamond is again a monoidal category under the \star product. Specifically, if AA and BB are \diamond-monoids, then the multiplication of ABA\star B is

(AB)(AB)(AA)(BB)AB. (A\star B) \diamond (A\star B) \to (A\diamond A) \star (B\diamond B) \to A\star B.

We define a bimonoid in CC to be a comonoid in the monoidal category (Mon (C),)(\Mon_\diamond(C),\star). An equivalent definition is obtained by considering \diamond-monoids in the category of \star-comonoids.

Of course, if CC is braided, this reduces to the usual definition of bimonoid in a braided monoidal category.

A nontrivially duoidal example is that a bimonoid with respect to the duoidal structure on functorial factorizations described above is precisely an algebraic weak factorization system.

Similarly, a duoid? is defined to be a \star-monoid in the monoidal category (Mon (C),)(\Mon_\diamond(C),\star) of \diamond-monoids, while a coduoid is a \diamond-comonoid in the category of \star-comonoids. Duoids, bimonoids, and coduoids in CC are equivalently duoidal functors 1C1\to C of the three possible kinds (see above).

Rings and near-rings

We can define (near-)rigs and near-rings in a duoidal category, where the additive and multiplicative monoidal structures are expressed with respect to \star and \diamond respectively. Since the distributive law requires duplication of variables, we need the additive monoidal structure to actually be a bimonoid (or a Hopf monoid in the case of a ring), and hence that \star is (on its own) braided.

More precisely, let CC be a duoidal category such that the tensor \star is braided (and probably some compatibility holds between the braiding and the duoidal structure). A near-rig in CC is of an object RR together with

  • A bimonoid structure on RR with respect to the braided monoidal category (C,,J)(C,\star,J), written as addition add:RRRadd:R\star R \to R and “coaddition” Δ:RRR\Delta:R\to R\star R;

  • A monoid structure on RR with respect to (C,,I))(C,\diamond,I)), written as multiplication mult:RRRmult : R \diamond R \to R;

  • The following diagram commutes, expressing right distributivity:

    (RR)R addid RR mult R idΔ add (RR)(RR) (RR)(RR) multmult RR \array{ (R\star R) \diamond R & \xrightarrow{add\diamond id} & R\diamond R & \xrightarrow{mult} & R\\ ^{id \diamond \Delta}\downarrow & & & & \uparrow^{add}\\ (R\star R) \diamond (R\star R) &\to & (R\diamond R) \star (R\diamond R) & \xrightarrow{mult \star mult} & R\star R }
  • The following diagram commutes, expressing right absorption:

    R ϵ J 0 R mult IR JR 0id RR \array{ R & \xrightarrow{\epsilon} & J & \xrightarrow{0} & R\\ ^\cong\downarrow & & & & \uparrow^{mult} \\ I\diamond R & \to & J \diamond R & \xrightarrow{0\diamond id} & R\diamond R }

It is a near-ring if the additive bimonoid structure is a Hopf monoid, and a rig or ring if the left distributive law and left absorption also hold.

Of course, this all makes sense in particular in a braided monoidal category regarded as duoidal. In particular, in a cartesian monoidal category, it reduces to the obvious internal definition of (near-)rigs and (near-)rings (since every object of a cartesian monoidal category is a comonoid in a unique way).

One can show that for near-rings, cocommutativity is automatic, in which case the Hopf monoid (R,add,Δ)(R,add,\Delta) can equivalently be regarded as a Hopf monoid in the cartesian monoidal category CComon (C)CComon_\star(C) of cocommutative comonoids in (C,,J)(C, \star, J). In this case, the above definitions coincide with the definitions of (near-)rings in the colax-distributive rig category induced by \diamond and the cartesian structure of CComon (C)CComon_\star(C). However, for general (near)-rigs, the definitions do not coincide, because cocommutativity is not automatic for near-rigs.

Strong and commutative monoids

Note that the constraints JJJJ\diamond J\to J and IJI\to J make JJ into a \diamond-monoid, and indeed also a bimonoid (since it is certainly a \star-comonoid). Also, just as the category of \diamond-monoids is \star-monoidal, the category of left, right, or bi-\diamond-modules over any bimonoid AA is \star-monoidal, with actions such as

(MN)A(MN)(AA)(MA)(NA)MN.(M\star N) \diamond A \to (M\star N) \diamond (A\star A) \to (M\diamond A) \star (N\diamond A) \to M\star N.

If CC admits reflexive coequalizers preserved in each variable by \diamond, then the category of (A,)(A,\diamond)-bimodules is in fact duoidal, with \diamond the usual tensor product of bimodules. (In the absence of such coequalizers, this category is virtually duoidal.)

We define a bistrong \star-monoid in a duoidal category to be a \star-monoid in the monoidal category of JJ-\diamond-bimodules. In the duoidal categories of endofunctors considered above, this reproduces the usual notion of “bistrong functor” (i.e. functor with a compatible left and right tensorial strength). Note that in a normal duoidal category, every object is uniquely a JJ-bimodule, so every \star-monoid is bistrong; and conversely, the duoidal category of (J,)(J,\diamond)-bimodules is (when it exists) always normal.

If the monoidal structure \diamond is braided (compatibly with \star), then a one-sided module automatically acquires a module structure on the other side. In this case, we define a strong \star-monoid to be a bistrong one whose JJ-\diamond-bimodule structure is obtained in this way.

Now let SS and TT be arbitrary objects and MM a (bi)strong \star-monoid. We say that two morphisms σ:SM\sigma:S\to M and τ:TM\tau:T\to M be morphisms commute if the following diagram commutes:

TS (TJ)(JS) (TJ)(JS) τσ (JT)(SJ) MM mult (JS)(TJ) στ MM mult M.\array{ T\diamond S & \xrightarrow{\cong} & (T\star J) \diamond (J \star S) & \to & (T\diamond J) \star (J\diamond S) \\ ^\cong\downarrow &&&& \downarrow^{\tau \star \sigma}\\ (J\star T) \diamond (S \star J) &&&& M\star M\\ \downarrow &&&& \downarrow^{mult}\\ (J\diamond S) \star (T\diamond J) & \xrightarrow{\sigma \star \tau} & M\star M & \xrightarrow{mult} & M. }

We say that MM is commutative if id Mid_M and id Mid_M commute.

In a braided monoidal category regarded as a normal duoidal one, this reduces to the usual notion of commutative monoid object. On the other hand, in the endofunctor-like examples above, it results in the notion of commutative monad or commutative theory.

Relation to intercategories

A duoidal category is the same as an intercategory with one object and only identity horizontal arrows, vertical arrows, horizontal cells, and vertical cells. The objects of the duoidal category are the basic cells of the intercategory, and its morphisms are the cubes. Like duoidal categories, intercategories support three (but not four) different kinds of morphism.


Last revised on April 28, 2017 at 17:49:35. See the history of this page for a list of all contributions to it.