nLab marked 2-limit

Motivation

In category theory, there is one sensible notion of limit, which is defined in terms of universal cones. However, when we move to enriched category theory, such limits (which we now call conical limits to distinguish them from more general notions of limits) are no longer sufficient. By “sufficient”, we mean here that there are notions in enriched category theory that are limit-like (such as powers) but that are not examples of conical limits. Furthermore, it is not true that the copresheaf construction exhibits the free completion under conical limits. This motivates the move to weighted limits, which are the appropriate notion of limit for enriched categories. In particular, the copresheaf construction is the free completion under weighted limits, and powers are examples of weighted limits.

For a general monoidal category $V$, weighted limits are more general than conical limits. However, when $V = Set$, every weighted limit can be expressed as a conical limit. This is useful, as the conical limits are simpler to describe than weighted limits.

A particular base of enrichment of interest is $Cat$, so that enriched categories are precisely 2-categories. The appropriate notion of limit for 2-categories is a weighted 2-limit, i.e. the usual notion of weighted limit for enriched categories (see 2-limit for more details). Unlike for $V = Set$, conical 2-limits do not suffice to capture all weighted 2-limits. Intuitively, cones only capture 1-cells, whereas it is also necessary to capture 2-cells for the appropriate notion of 2-limit.

However, it turns out that, just as every weighted limit for $V = Set$ can be reduced to a conical limit, every weighted limit for $V = Set$ can be reduced to a simpler kind of limit, called a marked limit. This is useful, as it makes working with 2-limits easier than working with general weighted limits.

History and terminology

While the notion of marked 2-limit goes back a long way, for one reason or another they have remained overlooked until recently.

• Marked 2-limits were introduced by John Gray in Adjointness for 2-Categories as cartesian quasi-limits.
• Ross Street proved the equivalence between marked 2-limits and weighted 2-limits.
• Szyld studied such 2-limits, calling them $\sigma$-s-limits.
• The terminology marked appears in Gagna–Harpaz–Lanari.
• A particular class of marked 2-limits, called lax normal conical 2-limits, (which suffices to capture all 2-limits) was studied by Mesiti.

Definition

Let $F : A \to B$ be a 2-functor between 2-categories, and let $A'$ be a locally full sub-2-category of $A$ (alternatively: a class of morphisms of $A$ closed under identities and composition). A marked-lax limit of $(A', F)$ is an object $m_l lim (A', F)$ together with a family of isomorphisms

$$B(b, m_l lim (A', F)) \cong [A, B]_{l, A'}(\Delta(b), F)$$

natural in $b \in B$, where $[A, B]_{l, A'}(\Delta(b), F)$ is the 2-category of 2-functors $A \to B$, marked-lax natural transformations (lax natural transformations $\alpha$ such that $\alpha_f = 1$ if $f \in A'$), and modifications.

Related pages

• 2-limit
• double limit
• F-category
• marked simplicial set

References

The concept is introduced in the following, where marked 2-limits are called cartesian quasi-limits:

• Adjointness for 2-Categories

It was generalised in section 0.2 of:

• Robert Blackwell?, Some existence theorems in the theory of doctrines, PhD thesis, UNSW Sydney, 1976.

The equivalence to weighted 2-limits is first proven in:

• Ross Street, Limits indexed by category-valued 2-functors Journal of Pure and Applied Algebra 8, Issue 2 (1976) pp 149-181. doi:10.1016/0022-4049(76)90013-X

• M.E. Descotte, Eduardo J. Dubuc, M. Szyld, Sigma limits in 2-categories and flat pseudofunctors, (v1: On the notion of flat 2-functors) arXiv:1610.09429 Adv. Math. 333 (2018) 266–313

• Martin Szyld. Lifting PIE limits with strict projections (2018), (arXiv:1809.04712)

• Luca Mesiti, The 2-Set-enriched Grothendieck construction and the lax normal conical 2-limits (2023), [arXiv:2302.04566]

On marked bilimits:

• Andrea Gagna, Yonatan Harpaz, and Edoardo Lanari, Bilimits are bifinal objects, Journal of Pure and Applied Algebra 226.12 (2022): 107137.

• Ivan Di Liberti, Axel Osmond, Bi-accessible and bipresentable 2-categories. [arXiv:2203.07046]

A proof of the equivalence between weighted 2-limits and marked 2-limits is given in the following, as well as a generalisation of these ideas to weighted limits for F-categories:

• Joanna Ko. Dotted Limits (2023), (arXiv:2306.01625)