nLab
duoidal category

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

Duoidal categories

Idea

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

Definition

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)

and

III I\to I\star I

while the laxity of JJ consists of transformations

JJJ J\diamond J \to J

and

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.

Examples

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

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.

References