nLab tight and loose morphisms

This page is about double categories. For the use of ‘tight’ and ‘loose’ in the context of F-categories, see there.

Idea

In the context of double category theory, vertical/horizontal or tight/loose are terminologies used to distinguish the two kinds of 1-cells comprising a double category.

Since a double category $\mathbb{D}$ is a category internal to $\mathbf{Cat}$, it comes with two kinds of 1-cells: the ones between objects of $\mathbb{D}_0$ (called tight) and the ones which are the objects of $\mathbb{D}_1$ (called loose). As explained in double category, loose cells are the protagonists in the specification of the internal category structure, since such structure tells us (1) what are their boundaries, (2) which one are the identities and (3) how they compose.

Historically, double categories were defined to be strict internal categories hence you could freely swap tight and loose directions without mathematical consequences (see (§3.2.2, §3.2.5, Grandis 2019). Recently, however, double categories are more commonly defined to be internal pseudocategories, meaning the category structure on the loose arrows is associative and unital only up to coherent isomorphism. As a consequence, loose cells do not organize in a category anymore, instead yielding a bicategory. Thus loose and tight cells are nowadays qualitatively different kinds of 1-cells and cannot be swapped (loose morphisms couldn’t form an object of $\mathbf{Cat}$).

The interplay between the weakness of loose cells and the strictness of tight cells is a major principle of double category theory (extending to multiple category theory too). Since the lack of strictness in the loose direction is described in terms of tight cells instead of other loose cells, double categories are tamer 2-dimensional structures than bicategories. For instance monoidal double categories (and other algebraic structures) are generally easier to define than monoidal bicategories (see refs here), or the fact lax double functors form a double category in a more natural way than lax 2-functors form a 2-category (see at icons).

Notation and terminology

Since 2-cells in double categories are naturally denoted by squares, their boundary 1-cells are predominantly referred to of ‘horizontal’ and ‘vertical’. This terminology is the most common in the published literature, albeit used inconsistently (i.e. sometimes vertical corresponds to loose, chiefly in Grandis and Parè’s papers, sometimes to tight). As discussed above, the modern usage of double categories doesn’t allow one to swap loose and tight cells freely, hence confusing vertical for horizontal is not just a nuisance but could make the difference between a true and a false theorem.

The terminology ‘tight’ and ‘loose’ comes from the world of F-categories, in particular from (Lack and Shulman, 2012). In that setting tight arrows are a special kind of loose arrows, but in the broader setting of double category theory that doesn’t have to be the case.

Relatedly, in a framed bicategory loose 1-cells are called ‘proarrows’ and tight 1-cells are called arrows. That’s because framed bicategories present proarrow equipments.

1-cells in $\mathbb{D}_0$	0-cells in $\mathbb{D}_1$
longitudinal	lateral	(Ehresmann 1963)
horizontal	vertical	Dawson and Paré; Grandis and Paré
vertical	horizontal	(Leinster 2004), (Cruttwell and Shulman, 2010)
tight	loose	(Lack and Shulman 2012)
dynamic	static	(Grandis 2019)
transversal	1-objects	(Grandis 2019)

Examples

1. In the double category $\mathbf{\mathbb{R}el}$ of sets, functions, relations and inclusions, functions are the tight cells and relations are the loose ones
2. In the double category $\mathbf{\mathbb{P}rof}$ of small categories, functors, profunctors and natural transformations, functors are the tight cells and profunctors are the loose ones

Loose bicategory and tight 2-category

From a (weak) double category $\mathbb{D}$ we can extract two separate `globular' objects:

1. The loose bicategory $L\mathbb{D}$, obtained by ignoring the tight morphisms: its objects are the same as $\mathbb{D}$, its 1-cells are the loose 1-cells of $\mathbb{D}$ and its 2-cells the squares. Horizontal composition and accompanying structure morphisms are inherited from the loose composition structure of $\mathbb{D}$, as well as the coherence laws. Vertical composition of 2-cells is given by tight composition of squares in $\mathbb{D}$, and thus satisfies interchange.
2. The tight 2-category $H\mathbb{D}$, obtained by ignoring the loose arrows: its objects are the same as $\mathbb{D}$, its 1-cells are the tight 1-cells of $\mathbb{D}$, its 2-cells are squares with trivial loose boundaries. Both horizontal and vertical compositions are defined by tightward composition of $\mathbb{D}$, hence are strict.

Related concepts

• double category
• framed bicategory

References

(Strict) double categories were introduced in

• Charles Ehresmann, Catégories double et catégories structurées, C.R. Acad. Paris 256 (1963) 1198-1201 [pdf, gallica]

where the two kinds of 1-cells were called longitudinal and lateral.

The less notationally-bound terminology tight/loose was introduced by

• Steve Lack, Mike Shulman, Enhanced 2-categories and limits for lax morphisms, Advances in Mathematics, vol. 229, 2012, (arxiv:1104.2111, doi:j.aim.2011.08.014)

Other references:

• Marco Grandis, Higher dimensional categories: from double to multiple categories, World Scientific, 2019. (doi:10.1142/11406)

• Geoffrey Cruttwell, Mike Shulman, A unified framework for generalized multicategories, TAC, vol. 24, 21, 2010, (arxiv:0907.2460), (tac)

• Tom Leinster, Higher operads, higher categories, London Math. Soc. Lec. Note Series 298, Cambridge University Press (2004) [math.CT/0305049, doi:10.1017/CBO9780511525896]