## Exact Completions and Toposes (2000)

Venue: | University of Edinburgh |

Citations: | 14 - 4 self |

### BibTeX

@TECHREPORT{Menni00exactcompletions,

author = {Matias Menni},

title = {Exact Completions and Toposes},

institution = {University of Edinburgh},

year = {2000}

}

### OpenURL

### Abstract

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to

### Citations

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275 | Foundation of Constructive Mathematics - Beeson - 1985 |

264 |
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Citation Context ...packages. 1.3 Prerequisites The basic examples use some recursion theory [97], topology [49] and the theory of locales [47]. Almost all the category theory involved can be found in Chapters I to V of =-=[70]-=-. As a convention, when we speak of a category (unqualified) we mean a locally small category. We may also omit to say that a category C has finite limits if the context makes this clear. On the other... |

199 |
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198 |
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196 | Toposes, Triples and Theories - Barr, Wells - 1985 |

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Citation Context ...on is expressed in terms of a connection between generic monos and generic proofs. Finally, there is a discussion of the relation of the characterization with work relating set theory and type theory =-=[2, 33]-=-. Chapter 1 --- Introduction 11 Chapters 6 and 8 can be seen as applications of the characterization, while Chapter 7 introduces the conceptual treatment of chaotic objects and proves some technical r... |

116 |
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Citation Context ...r words, E on X is Higgs-complete if for every equivalence relation D, the inclusion D ex (Y/D, X/E) # D ex/reg (Y/D, X/E) is actually an isomorphism. Compare also with the proposition in page 163 of =-=[58]-=- where it is shown that Cauchy-complete metric spaces enjoy, with respect to bimodules, a property similar to Higgs-completeness as in Definition 10.4.3. In order to relate Higgs-complete equivalence ... |

98 |
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79 |
spaces
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Citation Context ...For these, I acknowledge the use of Paul Taylor's useful diagrams and proof-tree packages. 1.3 Prerequisites The basic examples use some recursion theory [97], topology [49] and the theory of locales =-=[47]-=-. Almost all the category theory involved can be found in Chapters I to V of [70]. As a convention, when we speak of a category (unqualified) we mean a locally small category. We may also omit to say ... |

73 |
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Citation Context ...ally small category. We may also omit to say that a category C has finite limits if the context makes this clear. On the other hand, we refer the reader to [104] for the notion of a bi-adjoint and to =-=[31]-=- for the definition and results on factorization systems. For topos theory, Chapters I to VII of [71] should su#ce. Chapter 2 Regular, exact and lextensive categories In this chapter we present the ma... |

66 |
Realizability Toposes and Language Semantics
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Citation Context ...djoint to the global sections functor. Since then, realizability toposes have found diverse applications in logic and computer science, especially in the semantics of logics and programming languages =-=[40, 43, 100, 92, 66, 85, 84, 88, 68, 67, 65, 11, 12]-=-. In 1981, the regular completion and exact completion constructions for a category with finite limits appeared in C. Magno's thesis [72] (see also [19] and [16]). Chapter 1 --- Introduction 7 A few y... |

63 |
Elementary Categories, Elementary Toposes
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(Show Context)
Citation Context ...sk of finding new examples and counter-examples. In any case, such a su#cient condition was not looked for in [17, 32]. Related to this, it should also be mentioned that in 1995, McLarty presented in =-=[75]-=- necessary and su#cient conditions for an ex/reg completion to have power objects. But in order to check these conditions in practice one still has to build the ex/reg completion. So it was not possib... |

62 |
Abelian Categories, an Introduction to the Theory of Functors
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(Show Context)
Citation Context ...Regular, exact and lextensive categories The intuition is that an e#ective equivalence relation is one that has a "good" quotient. Let us mention two important classes of examples: abelian c=-=ategories [30]-=- (which we are not going to touch) and toposes. Topos theory (see [46, 71, 7, 75] and references therein) is a very rich theory. In this section we will only recall the most basic definitions. Definit... |

60 |
Locally Cartesian closed categories and type theory
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(Show Context)
Citation Context ...and the type theoretic versions of Russell's axiom of reducibility as explained in [2, 33]. In order to explain this we recall the interpretation of type theory in a locally cartesian closed category =-=[103]-=-. Given a locally cartesian closed category C, the idea is to interpret contexts as objects in C, types in context # as maps in C with codomain (the interpretation) of # and terms as sections of (the ... |

53 |
First steps in synthetic domain theory
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(Show Context)
Citation Context ...perty of ev # . Definition 7.6.3. A subsequential space X is extensional if ev # : X # # # X is a regular mono. It is pre-extensional if the map is a SSeq-pre-embedding. The terminology is taken from =-=[41]-=-. Now, recall that F : SSeq # Seq is the reflection functor. Lemma 7.6.4. If (ev # x i ) # ev # x in # # X then (x i ) # x in FX. Proof. Let O be sequentially open in X and x # O. It is clear that (O)... |

49 |
Sheaves and logic
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(Show Context)
Citation Context ...orical basis which will make them easier to understand. With the motivation of the thesis then understood, a second reading of the chronological perspective will be a lot clearer. In their 1979 paper =-=[27]-=-, Fourman and Scott presented the topos of sheaves on a locale as a category of non-standard equivalence relations. The essential idea behind this presentation is attributed to Higgs and referenced to... |

37 |
Some free constructions in realizability and proof theory
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- 1995
(Show Context)
Citation Context ...00, 92, 66, 85, 84, 88, 68, 67, 65, 11, 12]. In 1981, the regular completion and exact completion constructions for a category with finite limits appeared in C. Magno's thesis [72] (see also [19] and =-=[16]-=-). Chapter 1 --- Introduction 7 A few years later, in their 1988 paper [17], Carboni, Freyd and Scedrov showed that E# is the solution to a universal problem. This is a very nice and important concept... |

34 |
A reflection theorem for closed categories
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(Show Context)
Citation Context ...k [25] (recall that an epi map is called regular if it is the coequalizer of some pair of maps). These "imperfections" motivated research in order to find better categories of spaces (see fo=-=r example [24, 70, 38, 46, 59]-=- and more recently [102, 8, 21, 77, 78]). There is an obvious forgetful functor | | : Top # Set that assigns to each topological space its underlying set. This functor has both a left adjoint # and a ... |

31 | Handbook of categorical algebra. 2, volume 51 of Encyclopedia of Mathematics and its Applications - Borceux - 1994 |

31 |
An elementary theory of the category of sets
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(Show Context)
Citation Context ... history of topos theory and of regular and exact categories will not be discussed here. But in order to enter into the mood of this section let us mention some early references. For topos theory see =-=[4, 56, 57]-=- and for regular and exact categories see [6]. The contents of the section are divided in two. The first part is a chronological perspective on the events, results and problems that concern us in this... |

30 |
Proper maps of toposes
- Moerdijk, Vermeulen
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Citation Context ...tic situation. But this is the case in our examples. Between toposes, the geometric morphisms # # | | : C # S with # full and faithful and such that | | has a right adjoint # : S # C are called local =-=[48]-=- (see also [61, 63] where such strings of adjoints # # | | # # : S # C are called unity-and-identity-of-opposites and used to account for an abstract theory of space). Our purpose for introducing disc... |

27 |
Type theory via exact categories
- Birkedal, Carboni, et al.
- 1998
(Show Context)
Citation Context ... to show that the exact completion of Top (and also the exact completion of the full subcategory of T 0 topological spaces) is locally cartesian closed (see also [98]). Moreover, in their joint paper =-=[13]-=- with Birkedal and Scott, they used this result to show that Equ is locally cartesian closed by presenting it as the regular completion of (T 0 ) topological spaces. Also in 1998, Rosolini showed that... |

25 | Topological and limit-space subcategories of countablybased equilogical spaces
- Menni, Simpson
(Show Context)
Citation Context ...d regular if it is the coequalizer of some pair of maps). These "imperfections" motivated research in order to find better categories of spaces (see for example [24, 70, 38, 46, 59] and more=-= recently [102, 8, 21, 77, 78]-=-). There is an obvious forgetful functor | | : Top # Set that assigns to each topological space its underlying set. This functor has both a left adjoint # and a right adjoint #, both of which are full... |

25 |
The theory of triposes
- Pitts
- 1981
(Show Context)
Citation Context ...pos is denoted by Ass and its objects are called assemblies. These constructions by Higgs and Hyland motivated the invention of tripos theory. In 1980-81, tripos theory was presented in Pitts' thesis =-=[94]-=- and in his joint paper with Hyland and Johnstone [42]. Using tripos theory it is possible to treat uniformly the construction of toposes of sheaves on a locale and the construction of realizability t... |

25 |
Fibrations in bicategories, Cahiers de topologie et géométrie différentielle 21
- Street
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(Show Context)
Citation Context ...ak of a category (unqualified) we mean a locally small category. We may also omit to say that a category C has finite limits if the context makes this clear. On the other hand, we refer the reader to =-=[104]-=- for the notion of a bi-adjoint and to [31] for the definition and results on factorization systems. For topos theory, Chapters I to VII of [71] should su#ce. Chapter 2 Regular, exact and lextensive c... |

23 |
Locally Cartesian closed exact completions
- Carboni, Rosolini
(Show Context)
Citation Context ...a quasi-topos). In 1998, Carboni and Rosolini discovered a characterization of the categories Chapter 1 --- Introduction 9 with finite limits whose exact completions are locally cartesian closed (see =-=[21]-=-). This result can be used to show that the exact completion of Top (and also the exact completion of the full subcategory of T 0 topological spaces) is locally cartesian closed (see also [98]). Moreo... |

22 | On a topological topos - Johnstone - 1979 |

22 |
Qualitative Distinctions between some Toposes of Generalized Graphs
- Lawvere
- 1989
(Show Context)
Citation Context ...reflexive graphs. On the other hand, the presheaf topos Set # # is that of directed irreflexive graphs. By irreflexive here we mean that the graphs do not have a distinguished loop for each node. See =-=[60]-=- for more on these toposes. There are functors | | from each of these categories to the category of sets. As we have already mention in Chapter 2, in the cases of H+ , PAss, Ass, Top and SSeq the valu... |

21 | Developing theories of types and computability via realizability, Electronic Notes in Theoretical Computer Science 34, available at http://www.elsevier.nl/locate/entcs/volume34.html. The pdf version has active hyperreferences and is therefore the preferre
- Birkedal
(Show Context)
Citation Context ...djoint to the global sections functor. Since then, realizability toposes have found diverse applications in logic and computer science, especially in the semantics of logics and programming languages =-=[40, 43, 100, 92, 66, 85, 84, 88, 68, 67, 65, 11, 12]-=-. In 1981, the regular completion and exact completion constructions for a category with finite limits appeared in C. Magno's thesis [72] (see also [19] and [16]). Chapter 1 --- Introduction 7 A few y... |

21 |
Handbook of Categorical Algebra 1, volume 50 of Encyclopedia of Mathematics and its Applications
- Borceux
- 1994
(Show Context)
Citation Context ...lback along some f : Y # X. As | | is a left adjoint |e| is epi in S. As | | is a localization and epis are stable in S, |d| is epi. As | | is faithful it reflects epis and hence d is epi. Results in =-=[14]-=- (see Proposition 5.6.2) show that every localization | | # # induces a factorization system (E , M) in C characterized as follows. 1. f # E if and only if |f | is an isomorphism 2. f # M if and only ... |

20 |
On topological quotient maps preserved by pull-backs or products
- Day, Kelly
- 1970
(Show Context)
Citation Context ...tegory has exponentials. Regular epis in Top are not preserved by all functors X ( ) and hence Top is not cartesian closed ([44] and references therein) and regular epis are not stable under pullback =-=[25] (recall t-=-hat an epi map is called regular if it is the coequalizer of some pair of maps). These "imperfections" motivated research in order to find better categories of spaces (see for example [24, 7... |

19 |
The free exact category on a left exact one
- Carboni, Magno
- 1982
(Show Context)
Citation Context ...40, 43, 100, 92, 66, 85, 84, 88, 68, 67, 65, 11, 12]. In 1981, the regular completion and exact completion constructions for a category with finite limits appeared in C. Magno's thesis [72] (see also =-=[19]-=- and [16]). Chapter 1 --- Introduction 7 A few years later, in their 1988 paper [17], Carboni, Freyd and Scedrov showed that E# is the solution to a universal problem. This is a very nice and importan... |

18 |
A categorical approach to realizability and polymorphic types
- Carboni, Freyd, et al.
- 1987
(Show Context)
Citation Context ...ith a generic mono is a topos. Let us now compare this statement with other presentations of toposes that are ex/reg completions. The first one to mention is the presentation of the e#ective topos in =-=[17]-=- and [32]. They introduce the category of assemblies, build its ex/reg completion and show that it has power objects (which is an alternative way of defining a topos: a category with finite limits and... |

18 |
An abstract notion of realizability for which intuitionistic predicate calculus is complete
- Lauchli
- 1970
(Show Context)
Citation Context ...oolean presheaf toposes In this very short chapter we characterize the presheaf toposes that have a generic proof. We also explain briefly the connection of these toposes with Lauchli's realizability =-=[53, 62]-=-. Moreover, the characterization above will also let us find other examples of Grothendieck toposes whose exact completions are themselves toposes. 6.1 Boolean presheaf toposes In [16] the observation... |

17 |
Matching typed and untyped realizability
- Longley
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(Show Context)
Citation Context ...s examples and a good bibliography (see also [87]). For more on realizability toposes and applications of these ideas to the semantics of programming languages, logics and computation see for example =-=[40, 43, 100, 92, 66, 69, 85, 84, 88, 68, 67, 65, 11, 12]-=-. Although we are not going to be particularly interested in the class of pretoposess[74, 45] as such, we are going to encounter them (apart from dealing with toposes) from time to time, so we might a... |

16 |
The eective topos
- Hyland
- 1982
(Show Context)
Citation Context ...n a copy of this reference but it seems that it can be considered the starting point of the research reported here. This construction inspired that of realizability toposes. Indeed, in his 1982 paper =-=[39]-=-, Hyland attributes to Powell and Scott the idea of looking at realizability in a model theoretic way and uses this idea to explain the construction of the e#ective topos E# in a way resembling Higgs'... |

16 |
Quantifiers as sheaves
- Lawvere
- 1971
(Show Context)
Citation Context ... history of topos theory and of regular and exact categories will not be discussed here. But in order to enter into the mood of this section let us mention some early references. For topos theory see =-=[4, 56, 57]-=- and for regular and exact categories see [6]. The contents of the section are divided in two. The first part is a chronological perspective on the events, results and problems that concern us in this... |

16 |
Continuity and E#ectivity in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ...ussion of higher type recursion" [81] (see also [80]). It is the topos of sheaves for the canonical topology on the monoid of recursive functions. We now introduce the category GEn which was show=-=n in [99]-=- to be the category of separated objects for the double negation topology of the recursive topos. For this purpose, let R be the set of recursive functions and for any set S, let S N be the set of fun... |

15 |
Generalized Banach-Mazur functionals in the topos of recursive sets
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(Show Context)
Citation Context ... --- Regular, exact and lextensive categories 2.3.4 Generalized enumerated sets The recursive topos was introduced in [79] as a "suitable arena for discussion of higher type recursion" [81] =-=(see also [80]-=-). It is the topos of sheaves for the canonical topology on the monoid of recursive functions. We now introduce the category GEn which was shown in [99] to be the category of separated objects for the... |

13 |
Some thoughts on the future of category theory, Category Theory
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- 1990
(Show Context)
Citation Context ... of codiscrete objects. Related to this, it should be mentioned that Lawvere had already advocated for a conceptual use of codiscrete or chaotic objects in other areas of mathematics (see for example =-=[59, 55, 61, 63]-=-). It was very nice to contemplate these very di#erent classes of categories arising as solutions to the universal problems of finding regular, exact and ex/reg completions. But it was also clear that... |

13 |
Domain Theory in Realizability Toposes
- Phoa
- 1990
(Show Context)
Citation Context ...djoint to the global sections functor. Since then, realizability toposes have found diverse applications in logic and computer science, especially in the semantics of logics and programming languages =-=[40, 43, 100, 92, 66, 85, 84, 88, 68, 67, 65, 11, 12]-=-. In 1981, the regular completion and exact completion constructions for a category with finite limits appeared in C. Magno's thesis [72] (see also [19] and [16]). Chapter 1 --- Introduction 7 A few y... |

12 |
General function spaces, products and continuous lattices
- Isbell
- 1986
(Show Context)
Citation Context ...d if every object is exponentiable. In this case we may also say that the category has exponentials. Regular epis in Top are not preserved by all functors X ( ) and hence Top is not cartesian closed (=-=[44] and refer-=-ences therein) and regular epis are not stable under pullback [25] (recall that an epi map is called regular if it is the coequalizer of some pair of maps). These "imperfections" motivated r... |

11 |
Groups and Representations, volume 162 of Graduate Texts in Mathematics
- Alperin, Bell
- 1995
(Show Context)
Citation Context ... we only need to prove that 2 implies 3 and that 5 implies 2. In order to prove that 2 implies 3 consider first the case that C is a group. Then, Set C op is the topos of sets acted on by the group C =-=[45, 71, 3]-=-. It is well known (see [3] for example) that the connected objects in such a topos are the nonempty ones with only one orbit. Moreover, every connected object is isomorphic to one given by a coset sp... |

11 |
Cartesian closed exact completions
- Rosick´y
- 1999
(Show Context)
Citation Context ...ed (see [21]). This result can be used to show that the exact completion of Top (and also the exact completion of the full subcategory of T 0 topological spaces) is locally cartesian closed (see also =-=[98]-=-). Moreover, in their joint paper [13] with Birkedal and Scott, they used this result to show that Equ is locally cartesian closed by presenting it as the regular completion of (T 0 ) topological spac... |

10 |
Toposes generated by codiscrete objects in combinatorial topology and functional analysis. Notes for colloquium lectures given at North
- Lawvere
- 1989
(Show Context)
Citation Context ... of codiscrete objects. Related to this, it should be mentioned that Lawvere had already advocated for a conceptual use of codiscrete or chaotic objects in other areas of mathematics (see for example =-=[59, 55, 61, 63]-=-). It was very nice to contemplate these very di#erent classes of categories arising as solutions to the universal problems of finding regular, exact and ex/reg completions. But it was also clear that... |

9 |
Realizability models based on history-free strategies. Unpublished draft
- Abramsky, Longley
- 1999
(Show Context)
Citation Context ...a # . We call this PCA "Kleene's PCA" and we denote it by K 1 . There are many other examples of PCAs built using the theory of the lambda calculus, domain theory and, more recently, game th=-=eory (see [92, 66, 1]-=- for example) . It is possible to interpret a simple lambda calculus in any PCA and it is also possible to code easily data-types such as pairs or booleans. Because of this, we will write for example ... |

9 | Impredicativity entails untypedness
- Lietz, Streicher
- 1999
(Show Context)
Citation Context ...U . As explained in [68, 65] it can be shown that a TPCA has a universal type if and only if it is equivalent (in a suitable sense) to a PCA. The following result should be compared with Lemma 4.1 in =-=[65]-=- which is essentially the if direction. Proposition 8.3.4. A TPCA T has a universal type if and only if PAss(T ) has a generic object. Proof. For any type T let # T = (|T |, T, id |T | ). Consider fir... |

9 | A uniform account of domain theory in realizability models. To be submitted to special edition - Longley, Simpson - 1995 |