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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 6th 2012

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeMar 7th 2014
• (edited Mar 7th 2014)

Is there a systematic reference for this in a generalization in preadditive, or more generally, enriched categories ? I have in mind things like in this discussion but those references are not from the point of view of modality…

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeMar 11th 2014

Again, is there a worked out modal side/interpretation/theory/references for closure operations and reflections in enriched categories; in particular for $Ab$-enriched.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 11th 2014
• (edited Mar 11th 2014)

The punchline of modalities is that they are (co-)monads. So you would be looking for enriched monads (entry not existing yet, though…) such as strong monads (that exists!) and particularly additive monads.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeMar 11th 2014
• (edited Mar 11th 2014)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 13th 2014

Hi Zoran, I don’t have time to scan the literature list that you link to, but if you sbriefly say what you want to be understood under an additive Grothendieck topology, then I could see if I’d have a comment, or maybe some bystander here would.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeMar 14th 2014

Zoran, some buzzwords that you may already know include “nucleus” and “frames of torsion theories”. See for instance this review by Johnstone. In any case, it seems that various people have studied localizations in $Ab$-enriched contexts frame-theoretically (much as one can understand the collection of Grothendieck topologies as forming a frame – or is it a coframe?), as indicated in Johnstone’s review, and perhaps this would be relevant to your current interests.

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeMar 14th 2014
• (edited Mar 14th 2014)

Let $I_l R$ be the set of all left ideals in a ring $R$. It is naturally a preorder category with respect to the inclusion preorder This category is a lattice. For the localization questions another partial order $\succ$ on $I_l R$ is sometimes better: $K \succ J$ (category notation: $J \to K$) iff either $J \subset K$, or there exist a finite subset $w \subset R$ such that $(J:w) \subset K$. This order is important in ring theory; including in Rosenberg’s construction of spectra.

Any filter in $( I_l R, \succ )$ is called a uniform filter.

Notation: Given a left ideal $J \in I_l R$ and a subset $w \subset R$ define

Then $(J:w)$ is a left ideal in $R$. If $w =: K$ is also a left ideal, then $(J : K)$ is 2-sided ideal. In particular, if $w = K = R$, then $(J:R)$ is the maximal 2-sided ideal contained in $J$. For $r \in R$ we write $(J:r)$ for $(J,\{r\})$.

Given subsets $v,w \subset R$, set $((J:v):w)$ contains precisely all $t_1$ such that $t_1 w \subset (J:v)$, i.e. $t_1 w v \subset J$. Hence $((J:v):w) = (J: w v)$. $F$ is a filter of say left ideals:

(F1) $R \in F$ and $\emptyset \notin F$. (F2) If $J,K \in F$, then $J \cap K \in F$. (F3) If $J \in F$ and $J\subset K$ then $K \in F$.

A uniform filter (Pierre Gabriel calls it a topologizing filter) satisfies also

(UF) $J \in F \Leftrightarrow \, (\forall r \in R,\,(J:r) \in F)$.

It is in fact just a filter with a modified order on $I_l R$ as explained above.

A Gabriel filter of left ideals in a ring satisfies also

(GF) If $J \in F$ and $\forall j \in J$ the left ideal $(J':j)\in F$, then $J' \in F$.

To each uniform filter one attaches a localization functor, which has right adjoint in the GF case.

Now the sheafification functor for Grothendieck topologies is in complete analogy to the localization functor. The plus construction has an analogue in the noncommutative localization story, and if one squares it one gets the Gabriel localization functor. The story is rather complicated and seen more easily in a reformulation, see below.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeMar 14th 2014
• (edited Mar 14th 2014)

So, in the language of additive categories, quoting Daniel Murfet pdf

Let $A$ be a ringoid. A left additive topology on $A$ is a function $J$ assigning to each object $C$ a set $J(C)$ of left ideals at $C$, which satisfies

(i) The maximal left ideal $H_C$ ($Ab$-corepresentable) is always in $J(C)$.

(ii) If $a \in J(C)$ and $h : C\to D$, then $h_* a\in J(D)$.

(iii) If $a\in J(C)$ and $b$ is any left ideal at $C$, and if $f_* b \in J(D)$ for every $f:C\to D\in a$, then $b \in J(C)$.

This is in fact an analogue of cotopology with cosieves. Pushforward is defined by

If $b$ is a left ideal at $D$ then we define the pushout of b to be the following left ideal at $C$

$h_* b = \{ f : C\to X | f h \in b \}$

The dual form with pullbacks is suited for right ideals. In the case of additive category with one object the above definition is equivalent to the definition of Gabriel filter.

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeMar 14th 2014
• (edited Mar 14th 2014)

Now, the above theory is for $Ab$-enriched categories. But Rosenberg 1988 treatise from Stockholm has achieved both the Gabriel localization and the sheafification functor as special cases of sheafification for $Q$-categories; now there is no enriched Q-categories in his story, just a different ordinary $Q$-category for each case, non-enriched. Though I am not sure, maybe secretly one uses enriched Yoneda in doing presheaves in the first place, while constructing the Q-category of presheaves on original Q-category obtained from filter. Now, I would like to see what the modal side has to say. Closure operators are quite interesting on Gabriel side of the story.

And Todd, about the buzzwords, yes I heard of some and know of others, nuclei (pdf) are yet not comprehended by me even at most elementary level. There are several levels of lattices here. There are those in the story of construction of single localization, but also the lattice of all localizations. But those are not good for descent, as the pullback are too strong localizations and one instead needs iterated ones which are not pullbacks…this is long story which I will tell more once I get out of terrible double emergency I have the next 3-4 days.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeMar 24th 2014

I am now learning some things about nuclei from Pedro Resende. They give a class of closure operators which are in bijection with equivalence relations on locales/quantales when making the quotient locale/quantale. Among references one could mention Rosenthal’s 1990 book on quantales.