The Family Approach to Total Cocompleteness and Toposes

Street, Ross

doi:10.2307/1999291

Cited by 2 publications

(5 citation statements)

References 4 publications

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“…The following lemma and theorem are slight generalizations of results in [Str84] (see also [Kel91,Lemma 2.1]). 5.8.…”

Section: Definitionmentioning

confidence: 85%

“…On the one hand, Giraud's theorem characterizes categories of sheaves as the infinitary pretoposes with a small generating set. It is also folklore that adding disjoint universal coproducts is the natural "higher-ary" generalization of exactness; this is perhaps most explicit in [Str84]. Furthermore, the category of sheaves on a small infinitary-coherent category agrees with its infinitary-pretopos completion, as remarked in [Joh02].…”

Section: Introductionmentioning

confidence: 89%

“…In this section we define the notions of κ-ary regular and κ-ary exact categories, which are the outputs of our completion operations. These are essentially relativizations to κ of the notions of "familially regular" and "familially exact" from [Str84].…”

Section: Regularity and Exactnessmentioning

confidence: 99%

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Exact completions and small sheaves

Shulman

2012

Preprint

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We prove a general theorem which includes most notions of "exact completion" as special cases. The theorem is that "κ-ary exact categories" are a reflective sub-2-category of "κ-ary sites", for any regular cardinal κ. A κ-ary exact category is an exact category with disjoint and universal κ-small coproducts, and a κ-ary site is a site whose covering sieves are generated by κ-small families and which satisfies a solution-set condition for finite limits relative to κ.In the unary case, this includes the exact completions of a regular category, of a category with (weak) finite limits, and of a category with a factorization system. When κ = ω, it includes the pretopos completion of a coherent category. And when κ = K is the size of the universe, it includes the category of sheaves on a small site, and the category of small presheaves on a locally small and finitely complete category. The K-ary exact completion of a large nontrivial site gives a well-behaved "category of small sheaves".Along the way, we define a slightly generalized notion of "morphism of sites" and show that κ-ary sites are equivalent to a type of "enhanced allegory". This enables us to construct the exact completion in two ways, which can be regarded as decategorifications of "representable profunctors" (i.e. entire functional relations) and "anafunctors", respectively.

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“…The following lemma and theorem are slight generalizations of results in [Str84] (see also [Kel91,Lemma 2.1]). 5.8.…”

Section: Definitionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

Exact completions and small sheaves

Shulman

2012

Preprint

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“…As in the single epi case, any jointly regular epi family is automatically jointly strong epi. The converse is true when V is familialy regular, a proof of a stronger statement is given in [10].…”

Section: Appendix a Cauchy Completenessmentioning

confidence: 97%

“…On the other hand, the composition (10) for Y and Z equal X gives EpX, Xq `EpX, Xq ď EpX, Xq (18) (2) Adding EpX, Y q to the unit, and and the compositions…”

Section: Causal Spaces As Enriched Categoriesmentioning

confidence: 99%

Cauchy completeness and causal spaces

Nikolić

2017

Preprint

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Following Lawvere's description of metric spaces using enriched category theory, we introduce a change in the base of enrichment that allows description of some aspects of (relativistic) causal spaces. All such spaces are Cauchy complete, in the sense of enriched category theory. Furthermore, we give sufficient conditions on a base monoidal category for which enriched categories are Cauchy complete if and only if their underlying categories are (their idempotent arrows split). Contents 7 2.4. Cauchy completeness 7 3. Cauchy completeness via idempotent splitting 8 Appendix A. Cauchy completeness 13 Appendix B. Familial epiness 13 References 17

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The Family Approach to Total Cocompleteness and Toposes

Cited by 2 publications

References 4 publications

Exact completions and small sheaves

Exact completions and small sheaves

Cauchy completeness and causal spaces

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