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.