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arxiv: 2605.04516 · v2 · pith:D3Y3LJVGnew · submitted 2026-05-06 · 🧮 math.CT

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

Joanna Ko This is my paper

Pith reviewed 2026-06-30 23:46 UTC · model grok-4.3

classification 🧮 math.CT
keywords enhanced 2-categoriesmodels of sketches2-monadsalgebras over monadsw-rigged limitsorthogonal subcategory theoremenriched category theorylimit sketches
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The pith

For any enhanced limit 2-sketch with tight cones, the enhanced 2-category of its models in a locally presentable enhanced 2-category is equivalent to the 2-category of algebras over the induced enhanced 2-monad.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that models of an enhanced limit 2-sketch T with tight cones, taken in a locally presentable enhanced 2-category K, form an enhanced 2-category equivalent to the algebras over a certain enhanced 2-monad T. The equivalence identifies tight morphisms with strict algebra morphisms and loose morphisms with w-natural algebra morphisms. This yields a complete characterisation of limits in the model 2-category and shows that it inherits exactly the w-rigged limits of K. The argument also supplies an enriched form of the orthogonal subcategory theorem and extends monadicity and reflectivity results for sketch models to arbitrary locally presentable enriched bases.

Core claim

For any enhanced limit 2-sketch T with tight cones, the enhanced 2-category Mod_{s,w}(T, K) of models in a locally presentable enhanced 2-category K is equivalent to the enhanced 2-category T-Alg_{s,w} of algebras over the enhanced 2-monad T on Mod(T_τ, K), where the equivalence sends tight morphisms to strict T-morphisms and loose morphisms to w-T-morphisms. This equivalence completely characterises the limits that exist in Mod_{s,w}(T, K) and establishes that this 2-category inherits precisely all w-rigged limits.

What carries the argument

The enhanced 2-monad T induced on the 2-category of models of the tight part of the sketch, whose algebras recover the models of the full sketch.

Load-bearing premise

The ambient enhanced 2-category K must be locally presentable and the sketch T must be equipped with tight cones.

What would settle it

An explicit enhanced limit 2-sketch T with tight cones together with a non-locally presentable enhanced 2-category K in which the equivalence Mod_{s,w}(T, K) ≃ T-Alg_{s,w} fails to hold.

read the original 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 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that for any enhanced limit 2-sketch T equipped with tight cones, the enhanced 2-category Mod_{s,w}(T, K) of models in a locally presentable enhanced 2-category K (with tight morphisms the F-natural transformations and loose morphisms the w-natural transformations) is equivalent to the enhanced 2-category T-Alg_{s,w} of algebras over the induced enhanced 2-monad T on Mod(T_τ, K), with strict T-morphisms and w-T-morphisms. As a consequence, limits in Mod_{s,w}(T, K) are characterized and the category inherits precisely the w-rigged limits. The argument proceeds via an enriched analogue of the Orthogonal Subcategory Theorem together with a generalization of reflectivity and monadicity results for models of enriched limit sketches to arbitrary locally presentable enriched categories.

Significance. If the central equivalence and limit characterization hold, the work supplies a monadic description of models of enhanced 2-sketches that directly extends the classical limit-sketch/monad correspondence to the tight/loose enhanced 2-categorical setting. The enriched Orthogonal Subcategory Theorem and the monadicity results for arbitrary locally presentable enriched bases are reusable tools that strengthen the infrastructure for enriched and 2-categorical model theory.

minor comments (2)
  1. [Introduction] The abstract and introduction would benefit from an explicit pointer to the section containing the statement of the enriched Orthogonal Subcategory Theorem, so that readers can locate the new technical ingredient without scanning the entire development.
  2. [§2] Notation for the parameters s and w (strict and w-morphisms) is introduced in the abstract but receives its first full definition only later; a consolidated notation table or early paragraph would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main results, and recommendation to accept the manuscript. We are pleased that the central equivalence, the enriched Orthogonal Subcategory Theorem, and the monadicity results are viewed as significant and reusable contributions.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proves an equivalence Mod_{s,w}(T,K) ≃ T-Alg_{s,w} by first establishing an enriched Orthogonal Subcategory Theorem and then applying it to obtain reflectivity and monadicity for models of enhanced limit 2-sketches in locally presentable enriched 2-categories. These steps are stated as new results proved from the definitions of sketches, monads, and the enriched setting; the target equivalence is a direct consequence rather than an input. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. The hypotheses (K locally presentable as enhanced 2-category; T with tight cones) are external to the claimed result and match the conditions needed for the cited standard theorems to apply.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of 2-category theory, enriched categories, and the theory of sketches and monads; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard axioms and definitions of enhanced 2-categories, limit sketches, and 2-monads as developed in prior literature.
    Invoked throughout the abstract as the foundation for the equivalence and generalizations.
  • domain assumption Local presentability of the ambient enhanced 2-category K.
    Required for the model category to be well-behaved and for the monad to exist.

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