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I have expanded Lawvere-Tierney topology, also reorganized it in the process
In
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 ?
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 $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$.
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.
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).
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
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.
Oh, Urs, maybe you were not talking the short article in Gummersbach LNM volume (what I assumed)
but the later LNM volume
I can get that one too. I added the above bibliographic entries to Francis Borceux.
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.
Enrico Vitale was kind to send me the pdf scan of Borceux’s article from item 2 (Sheaves of algebras for a commutative theory).
@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: re enriched Grothendieck topologies, have you seen Borceux and Quinteiro, A theory of enriched sheaves, Cahiers 37(2), 1996?
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
They enrich over symmetric monoidal category.
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
Nice, thanks!
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.
Fixing a link to the now non-existent page ’historical note on Grothendieck topology’.
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
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