nLab weak morphism classifier

Idea

In 2-categorical algebra, there are many kinds of morphism between objects: e.g. strict morphisms, pseudo morphisms, lax morphisms, colax morphisms, etc. Typically it is the pseudo and (co)lax morphisms that arise in practice (for instance, we are rarely interested in strict monoidal functors), but the strict morphisms tend to be simpler to work with.

In many situations, we can study the weaker kinds of morphism using the strict kinds of morphism using a weak morphism classifier. A weak morphism classifier for an object AA is an object AA' such that weak morphisms ABA \to B are in natural bijection with strict morphisms ABA' \to B.

Dually, a weak morphism coclassifier is an object BB' such that weak morphisms ABA \to B are in natural bijection with strict morphisms ABA \to B'.

Definition

For algebras of 2-monads

Let TT be a 2-monad and denote by TAlg sT-Alg_s the 2-category of strict algebras? and strict morphisms for TT. Denote by TAlg wT-Alg_w the 2-category of strict algebras? and ww-weak morphisms for TT (where ww is pseudo, lax, colax, etc.). There is an identity-on-objects 2-functor:

TAlg sTAlg wT-Alg_s \to T-Alg_w

If this admits a left 2-adjoint ():TAlg wTAlg s(-)' : T-Alg_w \to T-Alg_s, we call this the ww-weak morphism classifier for TT. It if admits a right 2-adjoint TAlg wTAlg sT-Alg_w \to T-Alg_s, we call this the ww-weak morphism coclassifier for TT.

Weak morphism classifiers exist if and only if certain codescent objects exist in TAlg sT-Alg_s.

References

  • R. Blackwell, Max Kelly, John Power, Two-dimensional monad theory, (doi)
  • Stephen Lack, Codescent objects and coherence, Stephen Lack, J. Pure and Appl. Algebra 175 (2002) 223-241 doi
  • Stephen Lack and Michael Shulman, Enhanced 2-categories and limits for lax morphisms, Advances in Mathematics 229.1 (2012): 294-356.
  • Stephen Lack, Morita contexts as lax functors, Applied Categorical Structures 22.2 (2014): 311-330.
  • Kengo Hirata, Generalization of formal monad theory to lax functors, arXiv:2301.06420 (2023).

Last revised on October 3, 2024 at 18:04:21. See the history of this page for a list of all contributions to it.