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

[2605.04516] Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Category Theory arXiv:2605.04516 (math) [Submitted on 6 May 2026 ( v1 ), last revised 14 Aug 2026 (this version, v3)] Title: Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads Authors: Joanna Ko View a PDF of the paper titled Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads, by Joanna Ko View PDF 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 $\mathbb{M}\mathrm{od}(\mathcal{T}_\tau, \mathbb{K})$ 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. Comments: 38 pages Subjects: Category Theory (math.CT) MSC classes: 18C10, 18N15, 18C30, 18C40, 18D20 Cite as: arXiv:2605.04516 [math.CT] (or arXiv:2605.04516v3 [math.CT] for this version) https://doi.org/10.48550/arXiv.2605.04516 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Joanna Ko [ view email ] [v1] Wed, 6 May 2026 05:49:02 UTC (55 KB) [v2] Tue, 26 May 2026 21:53:23 UTC (56 KB) [v3] Fri, 14 Aug 2026 19:31:57 UTC (56 KB) Full-text links: Access Paper: View a PDF of the paper titled Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads, by Joanna Ko View PDF TeX Source view license Current browse context: math.CT < prev | next > new | recent | 2026-05 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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