PIE-limit in nLab

Contents

Contents

Idea

In 2-category-theory, by a PIE-limit one means a strict 2-limit which can be constructed from

1. (P) strict products,

2. (I) strict inserters,

3. (E) strict equifiers.

More precisely, the class of PIE-limits is the saturation of the class containing products, inserters, and equifiers. Any PIE-limit is in particular a flexible limit, and therefore also a (non-strict) 2-limit.

Furthermore, all strict pseudo-limits are PIE-limits, and therefore any strict 2-category which admits all PIE-limits also admits all non-strict 2-limits, although it may not have all strict 2-limits. This is the case, for instance, for the 2-category of strict algebras and pseudo morphisms over a strict 2-monad.

Some examples of PIE-limits are:

• Comma objects
• Pseudo limits
• Lax limits
• Oplax limits
• Eilenberg–Moore objects for monads
• Powers
• Iso-inserters
• Inverters
• Descent objects

An intuition is that PIE-limits are those 2-dimensional limits that do not impose any equations between 1-cells. For instance, equalizers and pullbacks are not PIE-limits.

PIE-limits can also be characterized as the coalgebras for a pseudo morphism classifier comonad, exhibiting them as a 2-categorical version of the notion of rigged limit.

Characterisation of PIE-weights

A PIE-limit is one whose weight is PIE. Power and Robinson characterised such weights W:J op→CatW : J^{op} \to Cat as those for which the induced functor ob∘W 0:J 0 op→Cat 0→Setob \circ W_0 \colon J^{op}_0 \to Cat_0 \to Set is multirepresentable. This holds if and only if each connected component of the category of elements of ob∘W 0ob \circ W_0 has an initial object.

• flexible limit
• semiflexible limit

References

• Blackwell, Kelly, and Power, Two-dimensional monad theory, Journal of Pure and Applied Algebra 59 (1989) 1-41. doi:10.1016/0022-4049(89)90160-6

• John Power, Edmund Robinson, A characterization of pie limits, Math. Proc. Cam. Phil. Soc. 110 (1991) 33 [doi:10.1017/S0305004100070092]

• John Bourke, Codescent objects in 2-dimensional universal algebra, PhD Thesis (2010), University of Sydney.

On λ\lambda-small objects in PIE-limits:

• Leonid Positselski, Notes on limits of accessible categories, arXiv:2310.16773.