Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads

Abstract

We establish the equivalence between models of enhanced $2$-sketches and algebras over monads, including the (co)lax morphisms. More precisely, for any enhanced limit $2$-sketch $\mathbb{T}$ with tight cones, the enhanced $2$-category $\mathbb{M}{\mathrm{od}}_{s,w}(\mathbb{T},\mathbb{K})$ of models of $\mathbb{T}$ in a locally presentable enhanced $2$-category $\mathbb{K}$, in which the tight and the loose morphisms are the $\mathcal{F}$-natural transformations and the loose $w$-natural transformations, respectively, is equivalent to the enhanced $2$-category ${\mathrm{T}\text{-}\mathbb{A}\lg }_{s,w}$ of algebras over an enhanced $2$-monad $T$ on the models $\mathbb{M}\mathrm{od}({\mathcal{T}}_{\tau },\mathbb{K})$ restricted to the tights with strict $T$-morphisms and $w$-$T$-morphisms. As a consequence, we explore the limits in the enhanced $2$-category $\mathbb{M}{\mathrm{od}}_{s,w}(\mathbb{T},\mathbb{K})$ of models with loose $w$-natural transformations, and conclude that $\mathbb{M}{\mathrm{od}}_{s,w}(\mathbb{T},\mathbb{K})$ inherits all $w$-rigged limits. Along the way, we establish an enriched analogue of the Orthogonal Sub-category Theorem, and generalise results on the reflectivity and the monadicity of models of enriched limit sketches in the base of enrichment to any arbitrary locally presentable enriched category.

38 pages