With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A lax monoidal category or skew monoidal category [Szlachányi 2012] is a monoidal category in which the associator- and unitor-transformations are not required to be invertible, i.e. are not required to be natural isomorphisms (as they are for ordinary monoidal categories).
In this case there needs to be made a choice in which direction these structure morphisms go. For the opposite of the “evident” direction one speaks of oplax monoidal categories.
For ordinary monoidal categories, the biased and unbiased definitions coincide up to equivalence (though this is a nontrivial coherence theorem), but in the lax and oplax cases this is no longer true. Moreover, in the biased cases we can make independent choices of the directions of various of the morphisms. This yields the following variations (in all cases we omit the coherence axioms for now):
An unbiased lax monoidal category has $n$-ary tensor products $(a_1\otimes \cdots\otimes a_n)$ for all $n\ge 0$, including a 0-ary unit $I = ()$, and generalized associativity maps such as
and a unit map $a\to (a)$. It is (strictly) normal if the later is an isomorphism (identity). It is a special case of a lax algebra for a 2-monad.
Dually, an unbiased oplax monoidal category has generalized associativity maps in the opposite direction
and a unit map $(a) \to a$. It is a special case of an oplax algebra for a 2-monad.
A left-biased lax monoidal category has a binary tensor product $a\otimes b$, a unit $I$, and constraints
A right-biased lax monoidal category has a binary tensor product $a\otimes b$, a unit $I$, and constraints
A left-biased oplax monoidal category has a binary tensor product $a\otimes b$, a unit $I$, and constraints
A right-biased oplax monoidal category has a binary tensor product $a\otimes b$, a unit $I$, and constraints
A left-skew monoidal category has a binary tensor product $a\otimes b$, a unit $I$, and constraints
A right-skew monoidal category has a binary tensor product $a\otimes b$, a unit $I$, and constraints
Terminologically, note that we use “right” and “left” to indicate the direction of the biased associator, while the direction of the unitors is indicated by “lax” or “oplax” (if they are in the same direction) or “skew” (if they are in different directions).
The relationship between these concepts is summarised in the following table.
If $V$ is a ___ monoidal category then… | $V^{rev}$ is… | $V^{op}$ is… | $V^{rev,op}$ is… |
---|---|---|---|
left-biased lax | right-biased lax | right-biased oplax | left-biased oplax |
right-biased lax | left-biased lax | left-biased oplax | right-biased oplax |
left-biased oplax | right-biased oplax | right-biased lax | left-biased lax |
right-biased oplax | left-biased oplax | left-biased lax | right-biased lax |
left-skew | right-skew | right-skew | left-skew |
right-skew | left-skew | left-skew | right-skew |
In other words, $rev$ and $op$ always swap direction; whilst $op$ interchanges lax and oplax (and has no effect on skew-monoidal categories).
There are two other possibilities, in which the definitions of left-skew and right-skew monoidal category are modified so that the associator is reversed, but they seem not to have been studied.
This use of “lax” and “oplax” in the biased case is justified by the fact that biased (op)lax monoidal categories are a special case of unbiased ones, with the $n$-ary tensor products defined in terms of the binary one by left- or right-associativity. The unbiased structures arising in this way can be characterized as those in which certain generalized associativities are identities (or, up to equivalence, isomorphisms).
The above definitions are complete except for the coherence axioms. In the unbiased case these can be deduced from the general notion of lax algebra. In the biased cases, the axioms are roughly the same as those in MacLane’s original definition of monoidal category: one pentagon, three unit triangles, and one unit-unit axiom. (Kelly’s later reduction of these to one pentagon and one unit triangle relies on invertibility of the constraints.) In all cases it is possible to orient all of these axioms so as to make sense.
(Lack and Street 2014, Theorem 8.1) The inclusion of right-normal left-skew monoidal categories (i.e. those for which the right unitor is invertible) and lax monoidal functors into left-skew monoidal categories admits a partial right pseudoadjoint, defined on those skew monoidal categories with reflexive coequalisers preserved by tensoring on the right, sending a skew monoidal category $C$ to the skew monoidal category $C^I$ of $I$-modules.
Consequently, any skew monoidal category with such coequalisers may be replaced by a right-normal skew monoidal category with the same monoids.
First note that all kinds of lax monoidal categories can be generalized to lax monoids in a monoidal bicategory. Using this generalization, we can make connections to many other kinds of monoidal structures:
Oplax monoidal categories (i.e. oplax monoids in Cat) can be identified with multicategories that are “weakly representable”.
Multicategories themselves are precisely lax monoids in Span, and also normal lax monoids in Prof.
We obtain various kinds of lax promonoidal category by working in Prof instead of Cat.
In particular, non-unital closed categories can be defined as biased lax monoids in Prof whose multiplication is corepresentable in the first argument and whose left unitor is invertible. If the identity is also representable, we obtain a (unital) closed category. The identification of biased lax monoids with particular unbiased ones thereby specializes to the identification of closed categories with closed multicategories.
Just as the delooping of a monoidal category is a bicategory, so too is the delooping of a skew-monoidal category a skew-bicategory.
The original definition (and first examples) of skew-monoidal categories:
and the dual notion of skew-closed categories:
See also:
Examples of skew-closed monoidal structures on model categories which become monoidal on the homotopy category:
Skew monoidal closed structures on the category of Gray categories are considered in:
Lax monoidal categories are called multitensors in:
Michael Batanin, Mark Weber: Algebras of higher operads as enriched categories, Applied Categorical Structures 19 (2011) 93-135 [arXiv:0803.3594]
Stephen Lack, Ross Street: On monads and warpings, Cahiers de topologie et géométrie différentielle 55 4 (2014) 244-266. [pdf]
Last revised on July 9, 2024 at 18:29:09. See the history of this page for a list of all contributions to it.