Completeness and cocompleteness of the categories of basic pairs and concrete spaces

Ishihara, Hajime; Kawai, Tatsuji

doi:10.1017/s0960129513000285

Cited by 6 publications

(15 citation statements)

References 14 publications

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“…We show that NID <ω is equivalent to the existence of equalisers in the category of concrete spaces, a predicative notion of point-set topology by Sambin [15]. Ishihara and Kawai [8,Section 4] have already shown that the category is complete and cocomplete using SGA. Hence, the essence of this subsection is that the converse holds.…”

Section: Equalisers In the Category Of Concrete Spacesmentioning

confidence: 83%

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Equivalents of the finitary non-deterministic inductive definitions

Hirata

Ishihara

Kawai

et al. 2019

Annals of Pure and Applied Logic

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We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the notion of biclosed subset. Our result suggests that proving these statements in constructive topology requires genuine extensions of CZF with the elementary or finitary NID.We work in a weak subsystem of CZF, called the elementary constructive set theory ECST [4], where none of the known fragments of the NID principle seems to be derivable.The language of ECST contains variables for sets and binary predicates = and ∈. The axioms and rules of ECST are the axioms and rules of intuitionistic predicate logic with equality, and the following set-theoretic axioms:Pairing: ∀a∀b∃y∀u (u ∈ y ↔ u = a ∨ u = b) .Union: ∀a∃y∀x (x ∈ y ↔ ∃u ∈ a (x ∈ u)) .

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Section: Equalisers In the Category Of Concrete Spacesmentioning

confidence: 83%

“…Remark 34. Ishihara and Kawai[8, Proposition 6.4] showed that CSpa has small products using SGA. Thus, if CSpa has equalisers, then CSpa is complete under FPA.…”

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confidence: 99%

Equivalents of the finitary non-deterministic inductive definitions

Hirata

Ishihara

Kawai

et al. 2019

Annals of Pure and Applied Logic

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“…As the final application, we show the following proposition which is the crucial step in the construction, given by Palmgren (2005), of coequalizers in the category of setpresented formal topologies; see Ishihara and Kawai (2014) for applications of our main result in the categories of basic pairs and concrete spaces introduced by Sambin (2003, forthcoming). (2005)).…”

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confidence: 83%

“…Finally, in Section 7, we will give some applications of the main result to algebra, topology and formal topology including some constructions mentioned above; see Ishihara and Kawai (2014) for applications in the categories of basic pairs and concrete spaces introduced by Sambin (2003, forthcoming).…”

Section: Introductionmentioning

confidence: 99%

Generalized geometric theories and set-generated classes

Aczel

Ishihara

Nemoto

et al. 2014

Math. Struct. Comp. Sci.

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We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is that the class of models of a generalized geometric theory is set-generated. Here, a class X of subsets of a set is set-generated if there exists a subset G of X such that for each α ∈ X , and finitely enumerable subset τ of α there exists a subset β ∈ G such that τ ⊆ β ⊆ α. We show the main result in the constructive Zermelo-Fraenkel set theory (CZF) with an additional axiom, called the set generation axiom which is derivable in CZF, both from the relativized dependent choice scheme and from a regular extension axiom. We give some applications of the main result to algebra, topology and formal topology.

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“…Remark 5.5 In [12], Ishihara and Kawai use non-deterministic inductive definitions to show that coequalizers exist in the categories of basic pairs and concrete spaces as introduced by Sambin [19,20].…”

Section: Applications To Formal Topologymentioning

confidence: 99%