Bibliographic record
Abstract
We show that morphisms from n homotopy unital A ∞ -algebras to a single one are maps over an operad module with n + 1 commuting actions of the operad A hu ∞ , whose algebras are homotopy unital A ∞ -algebras.The operad A ∞ and modules over it have two useful gradings related by isomorphisms which change the degree.The composition of A hu ∞ -morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures.The present work is a sequel to [Lyu15].We use freely notations and notions from the previous article.There polymodule cooperads were defined and an example of A ∞ -polymodule cooperad F was given.Here we describe three more examples of polymodule cooperads: an A ∞ -polymodule cooperad F, an A hu ∞ -polymodule cooperad F hu and A hu ∞ -polymodule cooperad F hu .Here A ∞ (resp.A hu ∞ ) is an operad of conventional (resp.homotopy unital) A ∞ -algebras, and A ∞ , A hu ∞ are their signless versions.Also F (resp.F hu ) is a signless version of F (resp.F hu ).We develop the idea of "isomorphism" of operads and polymodule cooperads changing degrees.Operads A ∞ and A ∞ , A hu ∞ and A hu ∞ , and polymodule cooperads F and F, F hu and F hu are "isomorphic" in this sense.Both categories of dg-operads and of polymodule dg-cooperads have a model structure.It is known that the dg-operad A ∞ (resp.A hu ∞ ) is a cofibrant resolution of the dg-operad As (resp.As1 ) of non-unital (resp.unital) associative dg-algebras.We show that the polymodule cooperad F (resp.F hu ) is a cofibrant resolution of the polymodule cooperad responsible for morphisms and composition in the multicategory of non-unital (resp.unital) associative dg-algebras.Polymodule cooperads F , F (resp.F hu , F hu ) are means to represent morphisms and their composition in multicategories of conventional (resp.homotopy unital) A ∞ -algebras or A ∞ -algebras.The composition is recovered via convolution of polymodule cooperad and a lax Cat-multifunctor Hom built from dg-modules.Verification that changing degrees does not lead out of polymodule cooperads is straightforward but lengthy.Contents 1 Preliminaries 1554 2 Model structure of the category of operad polymodules 1560
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".