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Let C be a finitely bicomplete category and W a subcategory. We prove that the existence of a model structure on C with W as the subcategory of weak equivalence is not first order expressible. Along the way we characterize all model structures where C is a partial order and show that these are determined by the homotopy categories.
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This content will become publicly available on August 1, 2026
Categories of hypermagmas, hypergroups, and related hyperstructures
In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas—which we call mosaics—form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory.
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- Award ID(s):
- 2201273
- PAR ID:
- 10597097
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Algebra
- Volume:
- 676
- ISSN:
- 0021-8693
- Page Range / eLocation ID:
- 408-474
- Subject(s) / Keyword(s):
- Canonical hypergroup category of hypergroups closed monoidal structure pointed simple matroid category of projective geometries
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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