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In 2-category-theory, by a PIE-limit one means a strict 2-limit which can be constructed from
(P) strict products,
(I) strict inserters,
(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:
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.
A PIE-limit is one whose weight is PIE. Power and Robinson characterised such weights as those for which the induced functor is multirepresentable. This holds if and only if each connected component of the category of elements of has an initial object.
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 -small objects in PIE-limits:
Last revised on June 12, 2024 at 09:35:43. See the history of this page for a list of all contributions to it.