Abstract
If $Bbb T$ is a semi-abelian algebraic theory, we prove that the category ${rm Born}^{Bbb T}$ of bornological $Bbb T$-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in ${rm Born}^{Bbb T}$ and, for a bornological $Bbb T$-algebra $X$, we investigate the relation between the representability of actions on $X$ as a $Bbb T$-algebra and as a bornological $Bbb T$-algebra. We investigate further the algebraic coherence and the algebraic local cartesian closedness of ${rm Born}^{Bbb T}$ and prove in particular that both properties hold in the case of bornological groups.
Highlights
Let C be a category with pullbacks
Given a semi-abelian algebraic theory T, the category BornT is homological in the sense of [6], and in particular, regular and protomodular
We investigate as well the algebraic coherence and the local algebraic cartesian closedness of BornT
Summary
Let C be a category with pullbacks. We consider its fibration of points (see [5]) whose stalk at an object X ∈ C has for objects the pairs s, p : A qqqqqqqqqqqqqqqqq X qqqqqqqqqqqqqqqqq with ps = idX. Local algebraic cartesian closedness is studied in [14], where it is shown in particular that the categories of groups and Lie algebras satisfy that property. In all these cases, it is proved that, with the notation above, it suffices to require the property in the case Y = 0, in which case the inverse image functor f ∗ is the kernel functor. Given a semi-abelian algebraic theory T, the category BornT is homological in the sense of [6], and in particular, regular and protomodular. We point out that local algebraic cartesian closedness of the category of bornological groups follows from Proposition 5.3 of [14] and our Proposition 2.7, but here we present an alternative proof
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