Bar Category of Modules and Homotopy Adjunction for Tensor Functors
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Bibliographic record
Abstract
Abstract Given a differentially graded (DG)-category ${{\mathcal{A}}}$, we introduce the bar category of modules ${\overline{\textbf{{Mod}}}-{\mathcal{A}}}$. It is a DG enhancement of the derived category $D({{\mathcal{A}}})$ of ${{\mathcal{A}}}$, which is isomorphic to the category of DG ${{\mathcal{A}}}$-modules with ${A_{\infty }}$-morphisms between them. However, it is defined intrinsically in the language of DG categories and requires no complex machinery or sign conventions of ${A_{\infty }}$-categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop a homotopy adjunction theory for tensor functors between derived categories of DG categories. It allows us to show in an enhanced setting that given a functor $F$ with left and right adjoints $L$ and $R$, the functorial complex $FR \xrightarrow{F{\operatorname{act}}{R}} FRFR \xrightarrow{FR{\operatorname{tr}} - {\operatorname{tr}}{FR}} FR \xrightarrow{{\operatorname{tr}}} {\operatorname{Id}}$ lifts to a canonical twisted complex whose convolution is the square of the spherical twist of $F$. We then write down four induced functorial Postnikov systems computing this convolution.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it