From Coherent Structures to Universal Properties

Details From Coherent Structures to Universal Properties From Coherent Structures to Universal Properties

2000-06-21

arxiv.org/abs/math/0006161v1

Mathematics

Category Theory

to appear in Journal of Pure and Applied Algebra

Scientific paper

Given a 2-category $\twocat{K}$ admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category $\twocat{L}$ with a 2-monad S on it such that: (1)S has the adjoint-pseudo-algebra property. (2)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 $\twocat{L}$ consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of $\twocat{K}$. We give an intrinsic characterisation of pseudo-S-algebras in terms of 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 monoid classifiers) as well as pseudo-functors into $\Cat$.

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