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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 18th 2011
   • (edited Oct 23rd 2012)
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   Author: Urs
   Format: MarkdownItexI have expanded [[Lawvere-Tierney topology]], also reorganized it in the process

   I have expanded Lawvere-Tierney topology, also reorganized it in the process

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2011
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   Author: zskoda
   Format: MarkdownItexIn * C. Cassidy, M. Hébert, [[G. M. Kelly]], _Reflective subcategories, localizations and factorization systems_, J. Austral. Math. Soc. Ser. A __38__ (1985), no. 3, 287–329, [MR86j:18001](http://www.ams.org/mathscinet-getitem?mr=779198); Corrigenda: "Reflective subcategories, localizations and factorization systems'', J. Austral. Math. Soc. Ser. A 41 (1986), no. 2, 286, [MR87f:18001](http://www.ams.org/mathscinet-getitem?mr=848041) there is somewhat strange wording "Grothendieck-Lawvere-Tierney topologies". He cites also some related articles to the topic of connection between topologies and localization, including a work of Borceux, which I can not find online (the MathReviews entry is quite useful though). Is there somebody who has the access ? * Francis Borceux, _Sheaves of algebras for a commutative theory_, Ann. Soc. Sci. Bruxelles Sér. I __95__ (1981), no. 1, 3–19, [MR83c:18006](http://www.ams.org/mathscinet-getitem?mr=628032)

   In

   • C. Cassidy, M. Hébert, G. M. Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. Ser. A 38 (1985), no. 3, 287–329, MR86j:18001; Corrigenda: “Reflective subcategories, localizations and factorization systems”, J. Austral. Math. Soc. Ser. A 41 (1986), no. 2, 286, MR87f:18001

   there is somewhat strange wording “Grothendieck-Lawvere-Tierney topologies”. He cites also some related articles to the topic of connection between topologies and localization, including a work of Borceux, which I can not find online (the MathReviews entry is quite useful though). Is there somebody who has the access ?

   • Francis Borceux, Sheaves of algebras for a commutative theory, Ann. Soc. Sci. Bruxelles Sér. I 95 (1981), no. 1, 3–19, MR83c:18006
3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2011
   • (edited May 18th 2011)
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   Author: zskoda
   Format: MarkdownItexBy localization C. Cassidy, M. Hébert, G. M. Kelly mean a more narrow sense of exact localization functor having right adjoint (i.e. a left exact reflection, they say "reflexion"). They say in the introduction > Localizations have been extensively studied for certain classes of categories $A$. When $A$ is the presheaf category $[K,Set]]$ they correspond to the Grothendieck topologies on $K$; and more generally when $A$ is any topos, they correspond to Grothendieck-Lawvere-Tierney topologies on $A$; see [9]. When $A$ is an additive functor category $[K,Ab]$, such as a category of modules, they correspond to Gabriel topologies, which are an additive analogue of the Grothendieck ones; see [13] and [14]. This has been further generalized by Borceux [3], replacing $Set$ and $Ab$ by the symmetric monoidal category of algebras fo any commutative Lawvere theory. Borceux [3] is the one quoted in reply 2 of this thread. The last two paragraphs of the MathReviews record [MR83c:18006](http://www.ams.org/mathscinet-getitem?mr=628032) for Borceux [3] (written by Harvey Wolf) says > To get a more tractable theory, the author restricts to the case where $\mathbf{T}$ is a commutative theory, in which case $\text{Set}^{\mathbf{T}}$ is a closed category. In this context, he considers a small $\text{Set}^{\mathbf {T}}$-category $\mathcal{C}$ and the $\text{Set}^{\mathbf{T}}$-category $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$. He defines a $\mathbf{T}$-sieve as a subfunctor of $\mathcal{C}(-,x)\colon\mathcal{C}^{\text{op}}\rightarrow\text{Set}^{\mathbf{T}}$. A $\mathbf{T}$-topology is a set $J(x)$ of $\mathbf{T}$-sieves for every $x$, satisfying the usual properties. He discusses the notion of a sheaf relative to such a topology and proves the existence and exactness of the associated sheaf functor. > The author shows, in the context of a commutative theory, that there is an object $\Omega_{\mathbf{T}}$ in $[\mathcal{C}^{\text{op}},\text{Set}]$ which classifies subobjects in $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$ and such that there exist bijections between the following: > (1) localizations of $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$; > (2) $\mathbf{T}$-topologies on $\mathcal{C}$; and > (3) morphisms $j\colon\Omega_{\mathbf{T}}\rightarrow\Omega_{\mathbf{T}}$ satisfying the Lawvere-Tierney axioms for a topology. > In the final section of the paper it is shown that for each localization $\mathcal{A}_j$ of $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$ there exists an object $\Omega_j$ in $[\mathcal{C}^{\text{op}},\text{Set}]$ which plays a role analogous to that played by $\Omega_{\mathbf{T}}$ for $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$.

   By localization C. Cassidy, M. Hébert, G. M. Kelly mean a more narrow sense of exact localization functor having right adjoint (i.e. a left exact reflection, they say “reflexion”). They say in the introduction

   Localizations have been extensively studied for certain classes of categories $A$. When $A$ is the presheaf category $[K,Set]]$ they correspond to the Grothendieck topologies on $K$; and more generally when $A$ is any topos, they correspond to Grothendieck-Lawvere-Tierney topologies on $A$; see [9]. When $A$ is an additive functor category $[K,Ab]$, such as a category of modules, they correspond to Gabriel topologies, which are an additive analogue of the Grothendieck ones; see [13] and [14]. This has been further generalized by Borceux [3], replacing $Set$ and $Ab$ by the symmetric monoidal category of algebras fo any commutative Lawvere theory.

   Borceux [3] is the one quoted in reply 2 of this thread. The last two paragraphs of the MathReviews record MR83c:18006 for Borceux [3] (written by Harvey Wolf) says

   To get a more tractable theory, the author restricts to the case where $\mathbf{T}$ is a commutative theory, in which case $\text{Set}^{\mathbf{T}}$ is a closed category. In this context, he considers a small $\text{Set}^{\mathbf {T}}$-category $\mathcal{C}$ and the $\text{Set}^{\mathbf{T}}$-category $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$. He defines a $\mathbf{T}$-sieve as a subfunctor of $\mathcal{C}(-,x)\colon\mathcal{C}^{\text{op}}\rightarrow\text{Set}^{\mathbf{T}}$. A $\mathbf{T}$-topology is a set $J(x)$ of $\mathbf{T}$-sieves for every $x$, satisfying the usual properties. He discusses the notion of a sheaf relative to such a topology and proves the existence and exactness of the associated sheaf functor.

   The author shows, in the context of a commutative theory, that there is an object $\Omega_{\mathbf{T}}$ in $[\mathcal{C}^{\text{op}},\text{Set}]$ which classifies subobjects in $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$ and such that there exist bijections between the following:

   (1) localizations of $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$;

   (2) $\mathbf{T}$-topologies on $\mathcal{C}$; and

   (3) morphisms $j\colon\Omega_{\mathbf{T}}\rightarrow\Omega_{\mathbf{T}}$ satisfying the Lawvere-Tierney axioms for a topology.

   In the final section of the paper it is shown that for each localization $\mathcal{A}_j$ of $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$ there exists an object $\Omega_j$ in $[\mathcal{C}^{\text{op}},\text{Set}]$ which plays a role analogous to that played by $\Omega_{\mathbf{T}}$ for $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2011
   • (edited May 18th 2011)
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   Author: zskoda
   Format: MarkdownItexI find very beautiful the generalization from the review quoted in (3). It is enriched version of Lawvere-Tierney topology, for a limited class of enrichments (over $Set^{\mathbf{T}}$, for a commutative theory $\mathbf{T}$ in $Set$). However, I do not have the access to the original paper. I added a paragraph into [[Lawvere-Tierney topology]]. I have to also understand the connection to Gabriel localization. The latter was the original motivation of Rosenberg to introduce Q-categories, and at some point he told me that the idea is related to the idea of Lawvere-Tierney in non-topos setup. I am sure he is not aware of the article by Borceux though.

   I find very beautiful the generalization from the review quoted in (3). It is enriched version of Lawvere-Tierney topology, for a limited class of enrichments (over $Set^{\mathbf{T}}$, for a commutative theory $\mathbf{T}$ in $Set$). However, I do not have the access to the original paper. I added a paragraph into Lawvere-Tierney topology.

   I have to also understand the connection to Gabriel localization. The latter was the original motivation of Rosenberg to introduce Q-categories, and at some point he told me that the idea is related to the idea of Lawvere-Tierney in non-topos setup. I am sure he is not aware of the article by Borceux though.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 18th 2011
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   Author: Urs
   Format: MarkdownItexThanks, Zoran, interesting. I tried to find that article by Borceux that you mention, but can't find it either. I can see a followup "Algebraic localization" but that's a Springer LNM volume and Springer is asking me to pay for it (I never quite understand why we are apparently not subscribed to such volumes online).

   Thanks, Zoran, interesting.

   I tried to find that article by Borceux that you mention, but can’t find it either. I can see a followup “Algebraic localization” but that’s a Springer LNM volume and Springer is asking me to pay for it (I never quite understand why we are apparently not subscribed to such volumes online).

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2011
   • (edited May 18th 2011)
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   Author: zskoda
   Format: MarkdownItexI would like that original paper, it seems more interesting for my purposes, than the other papers of Borceux from about the same time. I can get the LNM volume, though, and I have the * Francis Borceux, Gilberte Van Den Bossche, _Recovering a frame from its sheaves of algebras_, Journal of Pure and Applied Algebra __28__, 2, May 1983, 141-154, <a href="http://dx.doi.org/10.1016/0022-4049(83)90086-5">doi</a> P.S. The paper in the LNM Gummersbach volume has only short sketches of proofs. Supposedly there is a larger French preprint version with full proofs. * Francis Borceux, _Sur les localisations algébriques_, Preprint, Rapp. Sém. Math. Pures, Univ. Louvain, Louvain, Belgium, 1981

   I would like that original paper, it seems more interesting for my purposes, than the other papers of Borceux from about the same time. I can get the LNM volume, though, and I have the

   • Francis Borceux, Gilberte Van Den Bossche, Recovering a frame from its sheaves of algebras, Journal of Pure and Applied Algebra 28, 2, May 1983, 141-154, doi

   P.S. The paper in the LNM Gummersbach volume has only short sketches of proofs. Supposedly there is a larger French preprint version with full proofs.

   • Francis Borceux, Sur les localisations algébriques, Preprint, Rapp. Sém. Math. Pures, Univ. Louvain, Louvain, Belgium, 1981
7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2011
   • (edited May 18th 2011)
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   Author: zskoda
   Format: MarkdownItexOh, Urs, maybe you were not talking the short article in Gummersbach LNM volume (what I assumed) * Francis Borceux, _On algebraic localizations_, [MR84c:18007](http://www.ams.org/mathscinet-getitem?mr=682939), in Category theory (Gummersbach, 1981), pp. 13–22, Lecture Notes in Math. __962__, Springer 1982. but the later LNM volume * Francis Borceux, Gilberte Van den Bossche, _Algebra in a localic topos with applications to ring theory_, Lecture Notes in Mathematics __1038__, Springer-Verlag, Berlin, 1983. ix+240 pp. [MR85h:18005](http://www.ams.org/mathscinet-getitem?mr=724431) I can get that one too. I added the above bibliographic entries to [[Francis Borceux]].

   Oh, Urs, maybe you were not talking the short article in Gummersbach LNM volume (what I assumed)

   • Francis Borceux, On algebraic localizations, MR84c:18007, in Category theory (Gummersbach, 1981), pp. 13–22, Lecture Notes in Math. 962, Springer 1982.

   but the later LNM volume

   • Francis Borceux, Gilberte Van den Bossche, Algebra in a localic topos with applications to ring theory, Lecture Notes in Mathematics 1038, Springer-Verlag, Berlin, 1983. ix+240 pp. MR85h:18005

   I can get that one too. I added the above bibliographic entries to Francis Borceux.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2011
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   Author: zskoda
   Format: MarkdownItexBorceux-Bossche say that the correspondence between left exact localizations and Gabriel-Grothendieck topologies (that is an enriched version of Grothendieck topologies) fails once one gets to non-commutative theories. This was I think precisely the motivation for the introduction of sheaves on [[Q-categories]] rather than on Gabriel-Grothendieck sites in [[noncommutative algebraic geometry]].

   Borceux-Bossche say that the correspondence between left exact localizations and Gabriel-Grothendieck topologies (that is an enriched version of Grothendieck topologies) fails once one gets to non-commutative theories. This was I think precisely the motivation for the introduction of sheaves on Q-categories rather than on Gabriel-Grothendieck sites in noncommutative algebraic geometry.

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2011
   • (edited May 18th 2011)
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   Author: zskoda
   Format: MarkdownItexEnrico Vitale was kind to send me the pdf scan of Borceux's article from item 2 (Sheaves of algebras for a commutative theory).

   Enrico Vitale was kind to send me the pdf scan of Borceux’s article from item 2 (Sheaves of algebras for a commutative theory).

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2011
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    Author: DavidRoberts
    Format: MarkdownItex@Zoran #3 - I think 'reflexion' would be Max Kelly. I was in a talk once where he interjected about the distinction between 'dilation' and 'dilatation', quoting the Latin derivation to prove his point. :)

    @Zoran #3 - I think ’reflexion’ would be Max Kelly. I was in a talk once where he interjected about the distinction between ’dilation’ and ’dilatation’, quoting the Latin derivation to prove his point. :)

11. • CommentRowNumber11.
    • CommentAuthorFinnLawler
    • CommentTimeMay 19th 2011
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    Author: FinnLawler
    Format: MarkdownItexZoran: re enriched Grothendieck topologies, have you seen Borceux and Quinteiro, A theory of enriched sheaves, _Cahiers_ 37(2), 1996?

    Zoran: re enriched Grothendieck topologies, have you seen Borceux and Quinteiro, A theory of enriched sheaves, Cahiers 37(2), 1996?

12. • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeMay 19th 2011
    • (edited May 19th 2011)
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    Author: zskoda
    Format: MarkdownItexNo, I have not. Thanks. It is a bit dissappointing that those works with theories need commutative theories, what makes it strange that they mention Gabriel when they say Gabriel-Grothendieck topologies, as Gabriel's work on abelian localization is mainly made for (categories of modules over) noncommutative rings. So let me see the paper you suggest. Here is the full record now * Francis Borceux, Carmen Quinteiro, _A theory of enriched sheaves_, Cahiers de Topologie et Géométrie Différentielle Catégoriques __37__ no. 2 (1996), p. 145-162, [numdam](http://www.numdam.org/item?id=CTGDC_1996__37_2_145_0) They enrich over **symmetric** monoidal category.

    No, I have not. Thanks.

    It is a bit dissappointing that those works with theories need commutative theories, what makes it strange that they mention Gabriel when they say Gabriel-Grothendieck topologies, as Gabriel’s work on abelian localization is mainly made for (categories of modules over) noncommutative rings.

    So let me see the paper you suggest. Here is the full record now

    • Francis Borceux, Carmen Quinteiro, A theory of enriched sheaves, Cahiers de Topologie et Géométrie Différentielle Catégoriques 37 no. 2 (1996), p. 145-162, numdam

    They enrich over symmetric monoidal category.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2011
    • (edited Dec 5th 2011)
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    Author: Urs
    Format: MarkdownItexI am adding to [[Lawvere-Tierney topology]] a subsection [Properties -- Relation to lex reflectors](http://ncatlab.org/nlab/show/Lawvere-Tierney+topology#ClosureOperation). This is supposed to expose the baby-version of the $\sharp$-escape theory that Mike recently developed on the Café

    I am adding to Lawvere-Tierney topology a subsection Properties – Relation to lex reflectors. This is supposed to expose the baby-version of the $\sharp$-escape theory that Mike recently developed on the Café

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeDec 6th 2011
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    Author: Mike Shulman
    Format: MarkdownItexNice, thanks!

    Nice, thanks!

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexAdded to _[[Lawvere-Tierney operator]]_ links to [[geometric modality]] (currently redirecting to [[modal type theory]]) and added a pointer to Goldblatt's article to the References-section.

    Added to Lawvere-Tierney operator links to geometric modality (currently redirecting to modal type theory) and added a pointer to Goldblatt’s article to the References-section.

16. • CommentRowNumber16.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 20th 2021
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    Author: Richard Williamson
    Format: MarkdownItexFixing a link to the [now non-existent](https://nforum.ncatlab.org/discussion/12342/historical-note-on-grothendieck-topology/#Comment_89873) page 'historical note on Grothendieck topology'. <a href="https://ncatlab.org/nlab/revision/diff/Lawvere-Tierney+topology/44">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lawvere-Tierney+topology/44">v44</a>, <a href="https://ncatlab.org/nlab/show/Lawvere-Tierney+topology">current</a>

    Fixing a link to the now non-existent page ’historical note on Grothendieck topology’.

    diff, v44, current

17. • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeOct 11th 2023
    • (edited Oct 12th 2023)
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    Author: zskoda
    Format: MarkdownItexThe additive version of Lawvere-Tierney closure operators -- equivalent to [[Gabriel topology]] (=$Ab$-enriched Grothendieck topology) has been studied (and the equivalence proven) several years before Lawvere and Tierney in * [[Kim Lin Chew]], _Closure operations in the study of rings of quotients_, Bull. Math. Soc. Nanyang Univ. (1965) 1-20. <a href="https://ncatlab.org/nlab/revision/diff/Lawvere-Tierney+topology/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lawvere-Tierney+topology/47">v47</a>, <a href="https://ncatlab.org/nlab/show/Lawvere-Tierney+topology">current</a>

    The additive version of Lawvere-Tierney closure operators – equivalent to Gabriel topology (=$Ab$-enriched Grothendieck topology) has been studied (and the equivalence proven) several years before Lawvere and Tierney in

    • Kim Lin Chew, Closure operations in the study of rings of quotients, Bull. Math. Soc. Nanyang Univ. (1965) 1-20.

    diff, v47, current

18. • CommentRowNumber18.
    • CommentAuthornLab edit announcer
    • CommentTimeAug 1st 2024
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    Author: nLab edit announcer
    Format: MarkdownItexThere were some "#" instead of "\sharp" in the diagram Λ <a href="https://ncatlab.org/nlab/revision/diff/Lawvere-Tierney+topology/49">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lawvere-Tierney+topology/49">v49</a>, <a href="https://ncatlab.org/nlab/show/Lawvere-Tierney+topology">current</a>

    There were some “#” instead of “\sharp” in the diagram

    Λ

    diff, v49, current