# Codescent objects and coherence

### Stephen Lack

This appeared in Journal of Pure and
Applied Algebra 175:223-241, 2002.
We introduce 2-categorical colimit notions called codescent objects
of coherence data, and lax codescent objects of lax coherence data,
and use them to study the inclusion,
*T*-Alg_{s}-->Ps-*T*-Alg,
of the 2-category of strict *T*-algebras and strict *T*-morphisms
of a 2-monad *T* into the 2-category of pseudo *T*-algebras and
pseudo *T*-morphisms; and similarly the inclusion
*T*-Alg_{s}-->Lax-*T*-Alg_{l}, where
**Lax-***T*-Alg_{l} has lax algebras and lax morphisms rather
than pseudo ones. We give sufficient conditions under which these inclusions
have left adjoints. We give sufficient conditions under which the first
inclusion has left adjoint for which the components of the unit are
equivalences, so that every pseudo algebra is equivalent to a strict one.

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Steve Lack
Last modified: Fri Aug 30 09:11:18 EST 2002