Locally bounded enriched categories. (English) Zbl 1493.18001
Locally bounded categories subsume locally presentable categories and a significant number of other examples, including many topological categories and various elementary quasitoposes that are not locally presentable, as well as numerous categories of structures internal to these categories.
The authors of this paper achieve an important generalization of this concept, through two main objectives: one of them is the definition and study of the notion of a locally bounded enriched category as an object of study in its own right. The other is the achievement of a complete treatment of locally limited enriched categories that will help for future investigation of these enriched algebraic phenomena and results; generalizing their discoveries that locally bounded enriched categories provide a fruitful and more expansive environment for studying certain phenomena in enriched categorical algebra and, more generally, in the study of structures internal to enriched categories via enriched limit theories and related methods (see [R. B. B. Lucyshyn-Wright and J. Parker, “Presentations and algebraic colimits of enriched monads for a subcategory of arities”, Preprint, arXiv:2201.03466; “Diagrammatic presentations of enriched monads and varieties for a subcategory of arities”, in preparation; “Enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities”, in preparation]).
The authors of this paper achieve an important generalization of this concept, through two main objectives: one of them is the definition and study of the notion of a locally bounded enriched category as an object of study in its own right. The other is the achievement of a complete treatment of locally limited enriched categories that will help for future investigation of these enriched algebraic phenomena and results; generalizing their discoveries that locally bounded enriched categories provide a fruitful and more expansive environment for studying certain phenomena in enriched categorical algebra and, more generally, in the study of structures internal to enriched categories via enriched limit theories and related methods (see [R. B. B. Lucyshyn-Wright and J. Parker, “Presentations and algebraic colimits of enriched monads for a subcategory of arities”, Preprint, arXiv:2201.03466; “Diagrammatic presentations of enriched monads and varieties for a subcategory of arities”, in preparation; “Enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities”, in preparation]).
Reviewer: Joaquín Luna-Torres (Cartagena)
MSC:
| 18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |
| 18A32 | Factorization systems, substructures, quotient structures, congruences, amalgams |
| 18A35 | Categories admitting limits (complete categories), functors preserving limits, completions |
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 18B99 | Special categories |
| 18C10 | Theories (e.g., algebraic theories), structure, and semantics |
| 18C30 | Sketches and generalizations |
| 18C35 | Accessible and locally presentable categories |
| 18C40 | Structured objects in a category (group objects, etc.) |
| 18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
| 18D20 | Enriched categories (over closed or monoidal categories) |
| 18F60 | Categories of topological spaces and continuous mappings |
| 54B30 | Categorical methods in general topology |
Keywords:
locally bounded category; enriched category theory; closed category; locally presentable category; topological category; orthogonal subcategory; factorization system; concrete quasitopos; limit sketch; limit theoryReferences:
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