On toposes, algebraic theories, semi-abelian categories and compact Hausdorff spaces
Bibliographic record
Abstract
In this paper we study the categories C op * and (C op * ) T of T-models in C op * for an arbitrary algebraic theory T, when C is a topos or the category CHaus of compact Hausdorff spaces.It is well-known that, when C is a topos, C op * is semi-abelian.We show that CHaus op * is semi-abelian, as well as (CHaus op * ) T , and that, when C is a topos having locales of subobjects, (C op * ) T is also semi-abelian.In addition, we prove the representability of actions in CHaus op * .It is well-known that, given a topos E, the dual E op * of the category of pointed objects of E is semi-abelian (see [5]).We first prove that an analogous result holds in the context of compact Hausdorff spaces: the dual CHaus op * of the category of pointed compact Hausdorff spaces is semi-abelian.The Bourn-Janelidze characterization of semi-abelian algebraic theories (see [10]), and its generalization by Gran-Rosick (see [14]), indicate at once that adding arbitrarily operations (other than constants) and axioms to such a theory, one keeps a semi-abelian theory.This is what suggested us to investigate what occurs when adding arbitrary operations and axioms to E op * or CHaus op * , that is, when considering the categories (E op * ) T and (CHaus op * ) T of models of an arbitrary algebraic theory T in E op * or CHaus op * .And the answer is: all categories (CHaus op * ) T are semi-abelian.And, except for the existence of binary coproducts, the categories (E op * ) T satisfy all the other axioms for being semi-abelian, thus are homological (see [5]) and exact.And binary coproducts exist, thus (E op * ) T is semi-abelian, as soon as, in the topos E, the subobjects of every object constitute a locale.This is the case when E is a Grothendieck topos, but also when E is a topos of sheaves or presheaves of finite sets on a finite site.In a semi-abelian category, one has the notion of an object G acting on an object X, in terms of algebras for a certain monad.Actions on X are representable when the functor mapping G to the set of G-actions on X is representable.
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How this classification was reachedexpand
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 | 0.000 |
| 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 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".