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

Joanna Ko

Pith reviewed 2026-05-08 16:30 UTC · model grok-4.3

classification 🧮 math.CT

keywords enhanced 2-categoriessketches2-monadsmodels of sketchesorthogonal subcategory theoremw-rigged limitslocally presentable categories

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The pith

Models of any enhanced limit 2-sketch with tight cones are equivalent to algebras over an 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 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 a corresponding enhanced 2-monad. This equivalence accounts for both strict and loose morphisms, using F-natural transformations for tight ones and loose w-natural transformations for the others. The result lets one completely characterise limits in the model 2-category and shows it inherits exactly the w-rigged limits. It also establishes an enriched analogue of the Orthogonal Subcategory Theorem and extends reflectivity and monadicity results for models of enriched limit sketches to arbitrary locally presentable enriched categories.

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, in which the tight and loose morphisms are the F-natural transformations and the loose w-natural transformations respectively, is equivalent to the enhanced 2-category T-Alg_{s,w} of algebras over an enhanced 2-monad T on Mod(T_τ, K) restricted to tights with strict T-morphisms and w-T-morphisms. As a consequence the limits in Mod_{s,w}(T, K) are completely characterised and Mod_{s,w}(T, K) inherits precisely all w-rigged limits.

What carries the argument

The enhanced 2-monad T built on the 2-category of models of the tight sub-sketch T_τ, whose algebras recover the models of the full sketch T together with their tight and loose w-morphisms.

Load-bearing premise

The base K must be locally presentable as an enhanced 2-category and the sketch T must have tight cones.

What would settle it

An explicit enhanced limit 2-sketch with tight cones together with a locally presentable enhanced 2-category K in which the models of the sketch are not equivalent to the algebras over the induced 2-monad.

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

1 major / 2 minor

Summary. The paper establishes that for any enhanced limit 2-sketch T with tight cones, the enhanced 2-category Mod_{s,w}(T, K) of models (with tight F-natural transformations and loose w-natural transformations) in a locally presentable enhanced 2-category K is equivalent to the enhanced 2-category T-Alg_{s,w} of algebras over an enhanced 2-monad T on Mod(T_τ, K) (with strict T-morphisms and w-T-morphisms). As consequences, limits in Mod_{s,w}(T, K) are characterized and these 2-categories inherit all w-rigged limits. The work also proves an enriched analogue of the Orthogonal Subcategory Theorem and generalizes reflectivity and monadicity results for models of enriched limit sketches to arbitrary locally presentable enriched bases.

Significance. If the central equivalence holds, the result supplies a monadic description of models of enhanced 2-sketches that includes both tight and loose w-morphisms, extending prior enriched-category theorems. The generalization of the Orthogonal Subcategory Theorem to enriched bases and the explicit inheritance of w-rigged limits would be useful tools for higher categorical algebra and the study of sketches in 2-dimensional enriched settings.

major comments (1)

[Proof of the enriched Orthogonal Subcategory Theorem] The proof of the enriched Orthogonal Subcategory Theorem (used to establish reflectivity of the subcategory of models) lifts the 1-categorical factorization but does not contain an explicit verification that orthogonality holds for the loose w-natural transformations and their 2-cells. In particular, it is not shown that the mediating 2-cell is unique up to the w-structure or that the tight-cone condition interacts correctly with the loose enrichment. This step is load-bearing for the subsequent definition of the monad T on the full enhanced 2-category of models.

minor comments (2)

[Abstract] The abstract introduces dense notation (Mod_{s,w}, T-Alg_{s,w}, w-rigged limits) without a brief parenthetical gloss; a short clarifying sentence would improve accessibility.
[Consequences paragraph] The statement that Mod_{s,w}(T, K) 'inherits precisely all w-rigged limits' would benefit from a one-sentence reminder of the definition of w-rigged limits in the enhanced 2-categorical context.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this point in the proof of the enriched Orthogonal Subcategory Theorem. We address the comment below and will incorporate the suggested clarification in the revised version.

read point-by-point responses

Referee: [Proof of the enriched Orthogonal Subcategory Theorem] The proof of the enriched Orthogonal Subcategory Theorem (used to establish reflectivity of the subcategory of models) lifts the 1-categorical factorization but does not contain an explicit verification that orthogonality holds for the loose w-natural transformations and their 2-cells. In particular, it is not shown that the mediating 2-cell is unique up to the w-structure or that the tight-cone condition interacts correctly with the loose enrichment. This step is load-bearing for the subsequent definition of the monad T on the full enhanced 2-category of models.

Authors: We thank the referee for this observation. The proof in Section 4 proceeds by lifting the standard 1-categorical orthogonal factorization, with the enriched structure defined via the w-natural transformations and the tight-cone condition on the sketch. The orthogonality for loose w-morphisms and their 2-cells is ensured by the w-naturality axioms and the local presentability of the base K, which guarantees the existence and uniqueness of the mediating 2-cell up to the w-structure. Nevertheless, we agree that an explicit verification would strengthen the exposition. In the revised manuscript we will add a short dedicated paragraph immediately after the statement of the enriched Orthogonal Subcategory Theorem that records: (i) uniqueness of the mediating 2-cell follows from the w-naturality of the given transformations together with the orthogonality in the underlying 1-category, and (ii) the tight-cone condition is preserved because all cones in the sketch are required to be tight and the mediating morphism is constructed via the strict part of the factorization. This addition clarifies the interaction without altering the argument or the subsequent construction of the monad T. revision: yes

Circularity Check

0 steps flagged

No circularity: self-contained generalization of enriched orthogonality and monadicity theorems

full rationale

The paper proves an enriched analogue of the Orthogonal Subcategory Theorem in the locally presentable enriched setting and applies it to obtain reflectivity of the subcategory of models, followed by monadicity of the forgetful functor to establish the central equivalence Mod_{s,w}(T, K) ≃ T-Alg_{s,w}. These steps are derived from the stated assumptions on K and the sketch T rather than reducing any prediction or equivalence to a fitted parameter, self-definition, or load-bearing self-citation chain. The work generalizes prior 1-categorical and enriched results without renaming known patterns or smuggling ansatzes; the derivation chain remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions of enriched 2-category theory rather than new free parameters or invented entities.

axioms (2)

domain assumption K is a locally presentable enhanced 2-category
Explicitly required in the abstract for the model 2-category to be well-defined and for the equivalence to hold.
domain assumption The sketch T is an enhanced limit 2-sketch with tight cones
Stated as the hypothesis under which the monad T exists and the equivalence is proved.

pith-pipeline@v0.9.0 · 5599 in / 1430 out tokens · 44547 ms · 2026-05-08T16:30:00.979743+00:00 · methodology

discussion (0)

Reference graph

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