nLab 2-limits and 2-colimits in 2-categories of 2-algebras

Introduction

In moving from one dimension to two dimensions, there are a proliferation of concepts, with a choice of weakness for each one. For one, there is the notion of (strict) 2-category and bicategory, (strict) 2-monad and pseudomonad, and (strict) 2-algebras and pseudoalgebras. Therefore, while it is clear that “2-algebras for a 2-monad on a 2-category inherit 2-limits (and certain 2-colimits) from the base”, it can be tricky to recall which level of strictness is appropriate in each case. On this page, we list the various limit and colimit creation properties for two-dimensional algebras.

Limits

We shall use the conventional terminology of “2-” for strict concepts and “pseudo” for weak concepts to make it easier to compare with the references.

• The 2-category of 2-algebras and strict morphisms for a 2-monad on a 2-category inherits all 2-limits (this follows from $Cat$-enriched category theory).

• The 2-category of 2-algebras and pseudo morphisms for a 2-monad on a 2-category inherits all PIE 2-limits (§3 of BKP89.

• The 2-category of 2-algebras and lax morphisms for a 2-monad on a 2-category inherits all oplax limits, limits of strict morphisms, equifiers and inserters where one morphism is strict, Eilenberg–Moore objects for comonads, products and powers (see Lack05 and LS11). The page rigged limit contains more details.

• The 2-category of 2-algebras and pseudo morphisms for a flexible 2-monad? on a 2-category inherits all flexible limits (see Remark 7.2 of BKPS89).

• The 2-category of pseudoalgebras and pseudo morphisms for a pseudomonad on a 2-category inherits all bilimits (see Theorem 6.3.1.6 of Osmond21).

Related pages

• 2-monad

• colimits in categories of algebras

References

• R. Blackwell, G. M. Kelly, and A. J. Power, Two-dimensional monad theory, Jour. Pure Appl. Algebra 59 (1989), 1–41

• G. J. Bird, Max Kelly, John Power, Ross Street, Flexible limits for 2-categories, Journal of Pure and Applied Algebra 61 Issue 1 (1989) pp 1-27. doi:10.1016/0022-4049(89)90065-0

• Stephen Lack, Limits for lax morphisms, Appl. Categ. Structures,

  13(3):189–203, 2005

• Stephen Lack and Mike Shulman, Enhanced 2-categories and limits for lax morphisms, arXiv.

• Axel Osmond, A categorical study of spectral dualities, PhD thesis, Université Paris Cité, 2021.