Categories of continuous functors, I

Freyd, Peter; Kelly, G. M.

doi:10.1016/0022-4049(72)90001-1

Cited by 246 publications

(189 citation statements)

References 6 publications

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“…Most familiar is the notion of strong factorisation system (L, R) on a category C, introduced by Freyd and Kelly in [9]. This is given by two classes of maps L and R in C which are each closed under composition with isomorphisms, and which satisfy the axioms of (factorisation) Every map e : X → Y in C can be written as e = gf , where f ∈ L and g ∈ R; and (orthogonality) f ⊥ g for all f ∈ L and g ∈ R, where f ⊥ g means that for every commutative square…”

Section: 1mentioning

confidence: 99%

Understanding the Small Object Argument

Garner

2008

Appl Categor Struct

226

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The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

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Section: 1mentioning

confidence: 99%

Understanding the Small Object Argument

Garner

2008

Appl Categor Struct

226

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“…Existential m-variational doctrines P : C op G G InfSL are related to proper factorizations systems, see [HJ03]. Every such a doctrine determines a proper factorization system (E, M ) in C , see [FK72], where the monos in M are the comprehensions in C and the epis in E are surjective with respect to P , namely those arrows f :…”

Section: Proof a Weak Equalizer Ofmentioning

confidence: 99%

Triposes, exact completions, and Hilbert's ε-operator

Maietti¹,

Pasquali²,

Rosolini³

2017

Tbilisi Math. J.

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Triposes were introduced as presentations of toposes by J.M.E. Hyland, P.T. Johnstone and A.M. Pitts. They introduced a construction that, from a tripos P : C op G G Pos, produces an elementary topos TP in such a way that the fibration of the subobjects of the topos TP is freely obtained from P . One can also construct the "smallest" elementary doctrine made of subobjects which fully extends P , more precisely the free full comprehensive doctrine with comprehensive diagonals P cx : PrdP op G G Pos on P . The base category has finite limits and embeds into the topos TP via a functor K: PrdP G G TP determined by the universal property of P cx and which preserves finite limits. Hence it extends to an exact functor K ex : (PrdP ) ex/lex G G TP from the exact completion of PrdP .We characterize the triposes P for which the functor K ex is an equivalence as those P equipped with a so-called ε-operator. We also show that the tripos-to-topos construction need not preserve ε-operators by producing counterexamples from localic triposes constructed from wellordered sets. A characterization of the tripos-to-topos construction as a completion to an exact category is instrumental for the results in the paper and we derived it as a consequence of a more general characterization of an exact completion related to Lawvere's hyperdoctrines.

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“…The following concept was popularised by Peter Freyd and Max Kelly [89], but its origins can be traced back to Saunders Mac Lane [161] and John Isbell [121] (see also [23,Exercises 5.5] or [33, Section 5.5]).…”

Section: Propositionmentioning

confidence: 99%

Categorical Quantum Models and Logics

Heunen¹

2009

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Categories of continuous functors, I

Cited by 246 publications

References 6 publications

Understanding the Small Object Argument

Understanding the Small Object Argument

Triposes, exact completions, and Hilbert's ε-operator

Categorical Quantum Models and Logics

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