{"paper":{"title":"Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Models of any enhanced limit 2-sketch with tight cones are equivalent to algebras over an enhanced 2-monad.","cross_cats":[],"primary_cat":"math.CT","authors_text":"Joanna Ko","submitted_at":"2026-05-06T05:49:02Z","abstract_excerpt":"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}\\mat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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 an enhanced 2-monad T on Mod(T_τ, K), including tight and loose w-morphisms.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That K is locally presentable as an enhanced 2-category and that the sketch T has tight cones; the monadicity and limit characterizations rest on the base of enrichment being locally presentable and on the enriched Orthogonal Subcategory Theorem holding in this setting.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Models of enhanced limit 2-sketches are equivalent to algebras over enhanced 2-monads, including lax morphisms, and inherit w-rigged limits.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Models of any enhanced limit 2-sketch with tight cones are equivalent to algebras over an enhanced 2-monad.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"262a02491525d8e9425e52e9ca7eba5088cb92f45ef7b7bdfb20036e2e3cd064"},"source":{"id":"2605.04516","kind":"arxiv","version":2},"verdict":{"id":"65ae0bf3-a2a3-4433-8378-0cc93f325081","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T16:30:00.979743Z","strongest_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 an enhanced 2-monad T on Mod(T_τ, K), including tight and loose w-morphisms.","one_line_summary":"Models of enhanced limit 2-sketches are equivalent to algebras over enhanced 2-monads, including lax morphisms, and inherit w-rigged limits.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That K is locally presentable as an enhanced 2-category and that the sketch T has tight cones; the monadicity and limit characterizations rest on the base of enrichment being locally presentable and on the enriched Orthogonal Subcategory Theorem holding in this setting.","pith_extraction_headline":"Models of any enhanced limit 2-sketch with tight cones are equivalent to algebras over an enhanced 2-monad."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04516/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T11:40:47.174733Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.768264Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:21:39.928631Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"9a13013e6c37e277ba998dd2dcd86279e5e8f1410f4f443e68ff91d20bd5af85"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}