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am creating a table modalities, closure and reflection - contents and adding it as a floating table of contents to relevant entries
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…
Again, is there a worked out modal side/interpretation/theory/references for closure operations and reflections in enriched categories; in particular for $Ab$-enriched.
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.
Thanks for assurement. In fact, I expect that there is much more than just identifying enriched comonads in the story. You see there is a work on additive Gabriel-Grothendieck topologies, cf. literature in additive Grothendieck topology (zoranskoda). Localizations of specific type (not only reflective subcategories, but with some other exactness properties) correspond to Gabriel filters in various generalities. It is much beyond the known work of earlier work of H. Wolff on relation between enriched localizations, enriched isocounit adjunctions and related compatibility of ordinary localizations. It is also much beyond additive monads, as sufficient exactness makes monads automatically additive (from Eilenberg-Watts theorem) so this requirement seems not to give much. Now, if one would know some details on modal model theory and interpretations, one could spot interesting classes and so on. Thus I am looking for more concrete semantics, beyond singling out there is an enriched (co)monad in the game (I do sympathize with your creative ideas, which do infer so much math just from various adjunctions, the same attitude which Sasha Rosenberg had; just my need is more concrete now).
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.
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.
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.
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.
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.
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.
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