Algebraic exponentiation in general categories. (English) Zbl 1276.18002
Let \(\mathcal C\) be a category with pullbacks. The category \(\mathbf{Pt}(B)\) of points of an object \(B\) of \(\mathcal C\) is the comma category \((\mathrm{id}_B/(\mathcal C/B))\).
The category \(\mathcal C\) is locally Cartesian closed if and only if for every morphism \(p: E \to B\) of \(\mathcal C\), the induced pullback \(p^*: (\mathcal C/B) \to (\mathcal C/E)\) between the comma categories over \(B\) and \(E\), respectively, possesses a right adjoint. In this case, also the functor \(p^*: \mathbf{Pt}(E) \to \mathbf{Pt}(B)\) has a right adjoint, which is called an algebraic exponentiation in the paper under review.
In general, however, many categories of interest are not locally Cartesian closed. Thus, the question of whether at least an algebraic exponentiation in the above sense exists, becomes an interesting one. Several situations are studied by the author, in which this question can be answered and examples and counter-examples for the existence of such a right adjoint are given.
The category \(\mathcal C\) is locally Cartesian closed if and only if for every morphism \(p: E \to B\) of \(\mathcal C\), the induced pullback \(p^*: (\mathcal C/B) \to (\mathcal C/E)\) between the comma categories over \(B\) and \(E\), respectively, possesses a right adjoint. In this case, also the functor \(p^*: \mathbf{Pt}(E) \to \mathbf{Pt}(B)\) has a right adjoint, which is called an algebraic exponentiation in the paper under review.
In general, however, many categories of interest are not locally Cartesian closed. Thus, the question of whether at least an algebraic exponentiation in the above sense exists, becomes an interesting one. Several situations are studied by the author, in which this question can be answered and examples and counter-examples for the existence of such a right adjoint are given.
Reviewer: Marc Nieper-Wißkirchen (Augsburg)
MSC:
| 18A25 | Functor categories, comma categories |
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 18A22 | Special properties of functors (faithful, full, etc.) |
| 18A35 | Categories admitting limits (complete categories), functors preserving limits, completions |
| 18A30 | Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) |
| 18A05 | Definitions and generalizations in theory of categories |
| 18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |
| 18C20 | Eilenberg-Moore and Kleisli constructions for monads |
| 18C35 | Accessible and locally presentable categories |
| 18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
| 18D35 | Structured objects in a category (MSC2010) |
Keywords:
adjoint functor; split extension; internal action; pullback functor; semi-abelian category; protomodular category; (weakly) Mal’tsev category; (weakly) unital category; algebraic exponentiationReferences:
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