nLab lax monoidal category

Idea

A lax monoidal category is a monoidal category in which the associativity and unitality isomorphisms are replaced by not-necessarily-invertible transformations. An oplax monoidal category is similar except that the transformations go in the other direction.

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.

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

• Kornel Szlachányi, Skew-monoidal categories and bialgebroids, Advances in Mathematics 231.3-4 (2012): 1694-1730.

• John Bourke, Stephen Lack, Skew monoidal categories and skew multicategories, Journal of Algebra 506 (2018), 237-266. [arXiv:1708.06088, doi:10.1016/j.jalgebra.2018.02.039]

Skew monoidal closed structures on the category of Gray categories are considered in:

• John Bourke, Gabriele Lobbia, A skew approach to enrichment for Gray-categories, Advances in Mathematics volume 434 (2023). [arXiv:2212.12358, doi.org/10.1016/j.aim.2023.109327]

Lax monoidal categories are called multitensors in:

• Michael Batanin and Mark Weber, Algebras of higher operads as enriched categories, Applied Categorical Structures 19 (2011): 93-135.