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 $\mathscr{F}$-natural transformations and the loose $w$-natural transformations, respectively, is equivalent to the enhanced $2$-category ${\mathrm{T}\text{-}\mathbb{A}\mathrm{lg}}_{s, w}$ of algebras over an enhanced $2$-monad $T$ on the models $\mathbb{M}\mathrm{od}(\mathcal{T}_τ, \mathbb{K})$ restricted to the tights with strict $T$-morphisms and $w$-$T$-morphisms. As a consequence, we completely characterise 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 precisely 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.