nLab lax monoidal category

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Contents

Context

Monoidal categories

Idea
Overview of variations
Definitions
Normalisation
Related notions
References

Idea

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.

Overview of variations

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

$((a\otimes b\otimes c) \otimes () \otimes (d\otimes e)) \to (a\otimes b\otimes c\otimes d\otimes e)$

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

$(a\otimes b\otimes c\otimes d\otimes e) \to ((a\otimes b\otimes c) \otimes () \otimes (d\otimes e))$

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

$(a\otimes b)\otimes c\to a \otimes (b\otimes c) \qquad I \otimes a\to a \qquad a\otimes I\to a$
A right-biased lax monoidal category has a binary tensor product $a\otimes b$ , a unit $I$ , and constraints

$a \otimes (b\otimes c) \to (a\otimes b)\otimes c \qquad I \otimes a\to a \qquad a\otimes I\to a$
A left-biased oplax monoidal category has a binary tensor product $a\otimes b$ , a unit $I$ , and constraints

$(a\otimes b)\otimes c\to a \otimes (b\otimes c) \qquad a\to I \otimes a \qquad a\to a\otimes I$
A right-biased oplax monoidal category has a binary tensor product $a\otimes b$ , a unit $I$ , and constraints

$a \otimes (b\otimes c) \to (a\otimes b)\otimes c \qquad a\to I \otimes a \qquad a\to a\otimes I$
A left-skew monoidal category has a binary tensor product $a\otimes b$ , a unit $I$ , and constraints

$(a\otimes b)\otimes c\to a \otimes (b\otimes c) \qquad I \otimes a\to a \qquad a\to a\otimes I$
A right-skew monoidal category has a binary tensor product $a\otimes b$ , a unit $I$ , and constraints

$a \otimes (b\otimes c) \to (a\otimes b)\otimes c \qquad a\to I \otimes a \qquad a\otimes I\to a$

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).

Definitions

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.

Normalisation

Theorem

(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.

Related notions

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.

References

The definition of skew-monoidal category essentially appears in §2.2 of the following, but with only 2 of the 5 coherence axioms (i.e. Max Kelly‘s simplified axioms for monoidal categories, rather than Mac Lane’s original axioms):

Marek Zawadowski?, Lax monoidal fibrations, arXiv:0912.4464 (2009).

The notion was later rediscovered in Theorem 4 of:

Thorsten Altenkirch, James Chapman, and Tarmo Uustalu, Monads need not be endofunctors, Proceedings of the 13th international conference on Foundations of Software Science and Computational Structures (2010).

The definition was isolated in, and the terminology introduced in:

Kornél Szlachányi, Skew-monoidal categories and bialgebroids, Advances in Mathematics 231 3-4 (2012) 1694-1730 [arXiv:1201.4981, doi:10.1016/j.aim.2012.06.027]

and the dual notion of skew-closed categories:

Ross Street, Skew-closed categories, Journal of Pure and Applied Algebra 217 6 (2013) 973-988 [arXiv:1205.6522, doi:10.1016/j.jpaa.2012.09.020]