2-Category theory

Limits and colimits

A PIE-limit is a strict 2-limit which can be constructed from strict products, strict inserters, and 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.

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.


  • Blackwell, Kelly, and Power, Two-dimensional monad theory, Journal of Pure and Applied Algebra 59 (1989) 1-41

  • Power and Robinson, A characterization of pie limits, Math. Proc. Cam. Phil. Soc. (1991) 110, 33

Revised on July 4, 2016 00:09:31 by Mike Shulman (