# categories: coherence => universality (preprint)

The preprint "From coherent structures to universal properties"
is available from

http://www.maths.usyd.edu.au:8000/u/hermida

under coh-univ.ps

Abstract: Given a 2-category K admitting a calculus of bimodules, and a
2-monad T on it compatible with such calculus, we construct a 2-category
L with a 2-monad S on it such that: i) S has the adjoint-pseudo-algebra
property.

ii) The 2-categories of pseudo-algebras of S and T are equivalent.

Thus, coherent structures (pseudo-T-algebras) are transformed into
universally characterised ones (adjoint-pseudo-S-algebras). The
2-category L consists of lax algebras for the pseudo-monad induced by T
on the bicategory of bimodules of K. We give an intrinsic
characterisation of pseudo-S-algebras in terms of {\em
representability\/}. Two major consequences of the above transformation
are the classifications of lax and strong morphisms, with the attendant
coherence result for pseudo-algebras. We apply the theory in the context
of internal categories and examine monoidal and monoidal globular
categories (including their {\em monoid classifiers\/}) as well as
pseudo-functors into Cat.

--
Claudio Hermida

School of Mathematics and Statistics F07,
University of Sydney,
Sydney, NSW 2006,
Australia