On the uniqueness of cellular injectives

Rosický, Jiřı́

doi:10.1017/s0305004118000439

Cited by 4 publications

(2 citation statements)

References 23 publications

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“…Let us consider a category C that admits finite coproducts, binary pushouts when one leg is regular mono and such that regular monos are stable under pushous and composition. For example, these conditions are satisfied by every coregular category [34]. Following [1], we can construct a bicategory of cospans of C, denoted CoSpan(C): its objects are the same of C, while an arrow from X to Y is a pair of arrows…”

Section: Some Remarks On Cospansmentioning

confidence: 99%

Completeness and expressiveness for gs-monoidal categories

Fritz¹,

Gadducci²,

Trotta³

et al. 2022

Preprint

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Formalised in the study of symmetric monoidal categories, string diagrams are a graphical syntax that has found applications in many areas of Computer Science. Our work aims at systematising and expanding what could be thought of as the core of this visual formalism for dealing with relations and partial functions. To this end, we identify gs-monoidal categories and their graphical representation as a convenient, minimal structure that is useful to formally express such notions.More precisely, to show that such structures naturally arise, we prove that the Kleisli category of a strong commutative monad over a cartesian category is gs-monoidal. Then, we discuss how other categories providing a formalisation of "partial arrows", such as p-categories and restriction categories, are related to gs-monoidal categories. This naturally introduces a pre-order enrichment on gs-monoidal categories, and an equivalence of arrows, called "gs-equivalence": we conclude presenting a completeness result of this equivalence for models defined as lax functors to Rel.

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Section: Some Remarks On Cospansmentioning

confidence: 99%

Completeness and expressiveness for gs-monoidal categories

Fritz¹,

Gadducci²,

Trotta³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Minjectivity, for which M is a subclass of morphisms, is one of these generalizations which has been captured the interest of many mathematicians in different fields, [3,11,12,21]. Here we concentrate on another gener-alization of injectivity, that is, (M, E)-injectivity, which is also studied in different branches of mathematics, see for example [20]. Indeed, given two arbitrary classes of morphisms M and E in a category C, we say that an object Q is (M, E)-injective if any E-morphism f : A → Q can be extended through every M-morphism m : A → B.…”

Section: Introductionmentioning

confidence: 99%

(r,t)-injectivity in the category S-Act

Haddadi

Sheykholislami

2019

CGASA

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In this paper, we show that injectivity with respect to the class D of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category well-behaves. Indeed, if M is a subclass of monomorphisms, M ∩ D-injectivity well-behaves. We also introduce the notion of (r, t)-injectivity in the category S-Act, where r and t are Hoehnke radicals, and discuss whether this kind of injectivity well-behaves.

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Forking independence from the categorical point of view

Lieberman

Rosický

Vasey

2019

Advances in Mathematics

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Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we require a characterization suitable for work in µ-abstract elementary classes, i.e. accessible categories with all morphisms monomorphisms) and expository (we hope, with this account, to make forking accessible-and useful-to a broader mathematical audience). In particular, we present an axiomatic definition of what we call a stable independence notion on a category and show that this is in fact a purely category-theoretic axiomatization of the properties of model-theoretic forking in a stable first-order theory.

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On the uniqueness of cellular injectives

Cited by 4 publications

References 23 publications

Completeness and expressiveness for gs-monoidal categories

Completeness and expressiveness for gs-monoidal categories

(r,t)-injectivity in the category S-Act

Forking independence from the categorical point of view

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