Recently, in 22, it was shown that the category of cocommutative Hopf algebras over an arbitrary field k is semi-abelian. We extend this result to the category of cocommutative color Hopf algebras, ...i.e. of cocommutative Hopf monoids in the symmetric monoidal category of G-graded vector spaces with G an abelian group, given an arbitrary skew-symmetric bicharacter on G, when G is finitely generated and the characteristic of k is different from 2 (not needed if G is finite of odd cardinality). We also prove that this category is action representable and locally algebraically cartesian closed, then algebraically coherent. In particular, these results hold for the category of cocommutative super Hopf algebras by taking G=Z2. Furthermore, we prove that, under the same assumptions on G and k, the abelian category of abelian objects in the category of cocommutative color Hopf algebras is given by those cocommutative color Hopf algebras which are also commutative.
It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists in a distributive lattice of commuting congruences on A. The sheaf representations of ...universal algebras (over stably compact spaces) that arise in this manner have been recently characterised by Gehrke and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of the notion of softness.
In this paper, we extend the scope of this theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of “non-algebraic” examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames.
These results are used to construct sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).
Non-pointed abelian categories Duvieusart, Arnaud; Mantovani, Sandra; Montoli, Andrea
Theory and applications of categories,
01/2022, Volume:
38, Issue:
32
Journal Article
Peer reviewed
We study a property (P) of pushouts of regular epimorphisms along monomorphisms in a regular context. We prove that (P) characterizes abelian categories among homological ones. In the non-pointed ...case, we show that (P) implies the normality (in the sense of Bourn) of all subobjects, that any protomodular category satisfying (P) is naturally Mal'tsev, and that an exact category is penessentially affine if and only if it is protomodular and satisfies (P). An example of such a category is the one whose objects are the abelian extensions over an object in a (strongly) semi-abelian category; by exploiting some observations in this context, we also provide a characterization of strongly semi-abelian categories by a variant of the axiom of normality of unions.
To every regular category A equipped with a degree function δ one can attach a pseudo-abelian tensor category T(A,δ). We show that the generating objects of T decompose canonically as a direct sum. ...In this paper we calculate morphisms, compositions of morphisms and tensor products of the summands. As a special case we recover the original construction of Deligne’s category RepSt.
Propositions as [Types] Awodey, Steven; Bauer, Andrej
Journal of logic and computation,
08/2004, Volume:
14, Issue:
4
Journal Article
Peer reviewed
Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor A has turned up previously in a ...syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally Cartesian closed categories. We also show how to interpret first-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specifically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic first-order logic, in the sense that a formula from the fragment is derivable in intuitionistic first-order logic if, and only if, its interpretation in dependent type theory is inhabited. As a consequence, a modified double-negation translation into type theory (without bracket types) is complete, in the same sense, for all of classical first-order logic.
On linear exactness properties Jacqmin, Pierre-Alain; Janelidze, Zurab
Journal of algebra,
10/2021, Volume:
583
Journal Article
Peer reviewed
We study those exactness properties of a regular category C that can be expressed in the following form: for any diagram of a given ‘finite shape’ in C, a given canonical morphism between finite ...limits built from this diagram is a regular epimorphism. The main goal of the paper is to characterize essentially algebraic categories which satisfy this property via (essential versions of) linear Mal'tsev conditions, which are known to correspond to the so-called matrix properties. We then apply this characterization, along with our earlier work on preservation of exactness properties by pro-completions, to prove that these exactness properties can be reduced to matrix properties already in the general setting of regular categories.
We express the fundamental properties of commutative polygroups (also known as canonical hypergroups) in category-theoretic terms, over the category $ \mathbf{Set} $ formed by sets and functions. For ...this, we employ regularity as well as the monoidal structure induced on the category $ {\mathbf{Rel}} $ of sets and relations by cartesian products. We highlight how our approach can be generalised to any regular category. In addition, we consider the theory of partial multirings and find fully faithful functors between certain slice or coslice categories of the category of partial multirings and other categories formed by well-known mathematical structures and their morphisms.
The category of L-algebras Rump, Wolfgang
Theory and applications of categories,
01/2023, Volume:
39, Issue:
21
Journal Article
Peer reviewed
The category LAlg of L-algebras is shown to be complete and cocomplete, regular with a zero object and a projective generator, normal and subtractive, ideal determined, but not Barr-exact. ...Originating from algebraic logic, L-algebras arise in the theory of Garside groups, measure theory, functional analysis, and operator theory. It is shown that the category LAlg is far from protomodular, but it has natural semidirect products which have not been described in category-theoretic terms.
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