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Atrium > Mathematics, Physics & Philosophy: concrete objects in cohesive toposes

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeDec 28th 2010
- (edited Dec 28th 2010)
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Author: Urs
Format: MarkdownItexI imagine this thread to be a place where to put notes and have discussion about a question that David Carchedi and myself are interested in, several aspects of which we have been discussing here on various occasions already (related to diffeological and Frölicher spaces and their relation to topos theory). Here is a first part of this question: given a [[cohesive topos]] -- which implictly will mean [[cohesive (infinity,1)-topos]] -- $$ (\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : \mathbf{H} \to \infty Grpd $$ there ought to be a way to extract from just this general abstract data * a [[line object]] $\mathbb{A} \in \mathbf{H}$; * a [[coverage]] on the full subcategory of cartesian powers of the line object; * the structure of a [[pregeometry (for structured (infinity,1)-toposes)]] on this subcategory (or a slight variant thereof, where the pullback stability axiom of the admissible covers is replaced by a [[coverage]]-style stability condition). I imagine that one ought to pass to the full subcategory of $\mathbf{H}$ on those objects $X$ that are 1. 0-[[truncated]]; 1. in the kernel of $\Pi$ (= geometrically contractible): $\Pi(X) \simeq *$ 1. concrete, in that $X \to Codisc \Gamma X$ is a monomorphism (a (-1)-truncated morphism). This category is closed under finite products in $\mathbf{H}$. I want to say that it can be equipped with a coverage where $\{U_i \to U\}$ is covering if $\coprod_i U_i \to U$ is an effective epimorphism in $\mathbf{H}$. The problem to be solved then is to show (or not) that for any morphism $f : V \to U$ we can fill diagrams $$ \array{ V_i &\to& U_I \\ \downarrow && \downarrow \\ V &\stackrel{f}{\to}& U } $$ such that $\{V_i \to V\}$ is covering. The strategy ought to be 1. form the pullback $ \array{ f^* U_i &\to& U_I \\ \downarrow && \downarrow \\ V &\stackrel{f}{\to}& U } $ and show that $f^* U_i$ itself can be covered by $\{V_{i,j} \to f^* U_i\}$. 1. To do so, show that 0-truncated concrete objects are "topological spaces equipped with geometric structure" as we know is the case for the case that $\mathbf{H}$ is sheaves on [[CartSp]] (where these are the [[diffeological space]]). And then use general statements about refinements of open covers by [[good open cover]]s. I didn't really have much of a chance to think about this over the holidays, and so at this moment I have little insight on what sounds like a simple question: **Question:** How can we canonically associate a [[locale]], to a 0-truncated concrete object $X$ in a general cohesive topos $\mathbf{H}$? (Notice that we can always form the homotopy type $\Pi(X)$. But here I want an actual locale/topological space that realizes this homotopy type, associated naturally to $X$.) A first idea might be to look at the over-topos $\mathbf{H}/X$ and see if the fact that $X$ is 0-truncated and concrete implies that there is a localic $\infty$-topos to be obtained from this. Another idea might be to look at the poset of subobjects of $X$ in $\mathbf{H}$ and see if the fact that $X$ is 0-truncated and concrete implies that there is a locale to be obtained from this. I imagine the general strategy would be to show this for the case that we actually have equality $X = Codisc \Gamma X$ and then deduce that the corresponding property extends along monomorphisms $X \hookrightarrow Codisc \Gamma X$. I played around with these ideas a bit, but not really successfully. I think I am just being dense, since tis smells like it ought to have a simple answer. So for the moment I just leave this note here.
I imagine this thread to be a place where to put notes and have discussion about a question that David Carchedi and myself are interested in, several aspects of which we have been discussing here on various occasions already (related to diffeological and Frölicher spaces and their relation to topos theory).

Here is a first part of this question:

given a cohesive topos – which implictly will mean cohesive (infinity,1)-topos –
(Π⊣Disc⊣Γ⊣Codisc):H→∞Grpd (\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : \mathbf{H} \to \infty Grpd
there ought to be a way to extract from just this general abstract data
- a line object $\mathbb{A} \in \mathbf{H}$ ;
- a coverage on the full subcategory of cartesian powers of the line object;
- the structure of a pregeometry (for structured (infinity,1)-toposes) on this subcategory (or a slight variant thereof, where the pullback stability axiom of the admissible covers is replaced by a coverage-style stability condition).
I imagine that one ought to pass to the full subcategory of $\mathbf{H}$ on those objects $X$ that are
1. 0-truncated;
2. in the kernel of $\Pi$ (= geometrically contractible): $\Pi(X) \simeq *$
3. concrete, in that $X \to Codisc \Gamma X$ is a monomorphism (a (-1)-truncated morphism).
This category is closed under finite products in $\mathbf{H}$ . I want to say that it can be equipped with a coverage where $\{U_i \to U\}$ is covering if $\coprod_i U_i \to U$ is an effective epimorphism in $\mathbf{H}$ .

The problem to be solved then is to show (or not) that for any morphism $f : V \to U$ we can fill diagrams
V i → U I ↓ ↓ V →f U \array{ V_i &\to& U_I \\ \downarrow && \downarrow \\ V &\stackrel{f}{\to}& U }
such that $\{V_i \to V\}$ is covering. The strategy ought to be
1. form the pullback $\array{ f^* U_i &\to& U_I \\ \downarrow && \downarrow \\ V &\stackrel{f}{\to}& U }$ and show that $f^* U_i$ itself can be covered by $\{V_{i,j} \to f^* U_i\}$ .
2. To do so, show that 0-truncated concrete objects are “topological spaces equipped with geometric structure” as we know is the case for the case that $\mathbf{H}$ is sheaves on CartSp (where these are the diffeological space). And then use general statements about refinements of open covers by good open covers.
I didn’t really have much of a chance to think about this over the holidays, and so at this moment I have little insight on what sounds like a simple question:

Question: How can we canonically associate a locale, to a 0-truncated concrete object $X$ in a general cohesive topos $\mathbf{H}$ ?

(Notice that we can always form the homotopy type $\Pi(X)$ . But here I want an actual locale/topological space that realizes this homotopy type, associated naturally to $X$ .)

A first idea might be to look at the over-topos $\mathbf{H}/X$ and see if the fact that $X$ is 0-truncated and concrete implies that there is a localic $\infty$ -topos to be obtained from this.

Another idea might be to look at the poset of subobjects of $X$ in $\mathbf{H}$ and see if the fact that $X$ is 0-truncated and concrete implies that there is a locale to be obtained from this.

I imagine the general strategy would be to show this for the case that we actually have equality $X = Codisc \Gamma X$ and then deduce that the corresponding property extends along monomorphisms $X \hookrightarrow Codisc \Gamma X$ .

I played around with these ideas a bit, but not really successfully. I think I am just being dense, since tis smells like it ought to have a simple answer. So for the moment I just leave this note here.
- CommentRowNumber2.
- CommentAuthorMike Shulman
- CommentTimeDec 29th 2010
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Author: Mike Shulman
Format: MarkdownItexWell, any (Grothendieck) topos has a localic reflection (via the hyperconnected-localic factorization), and in particular that applies to $\mathbf{H}/X$, with no need for truncation or codiscreteness. The localic reflection of a topos is the locale corresponding to its poset of subterminal objects, which is a frame, and for $\mathbf{H}/X$ that would be the subobjects of X.

Well, any (Grothendieck) topos has a localic reflection (via the hyperconnected-localic factorization), and in particular that applies to $\mathbf{H}/X$ , with no need for truncation or codiscreteness. The localic reflection of a topos is the locale corresponding to its poset of subterminal objects, which is a frame, and for $\mathbf{H}/X$ that would be the subobjects of X.
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeDec 29th 2010
- PermaLink
Author: Urs
Format: MarkdownItexRight, thanks. I should have formulated the question better: the task is to identify a topological space/locale underlying a concrete object $X$ that has the homotopy type of $\Pi(X)$ and realizes $X$ as a topological space equipped with extra structure. More later, have to dash off now.

Right, thanks. I should have formulated the question better: the task is to identify a topological space/locale underlying a concrete object $X$ that has the homotopy type of $\Pi(X)$ and realizes $X$ as a topological space equipped with extra structure. More later, have to dash off now.
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeDec 29th 2010
- (edited Dec 29th 2010)
- PermaLink
Author: Urs
Format: MarkdownItexFor the time being, I am coming back to seeing how concrete objects may serve to present a canonical _geometry_ (in the sense of: site with compatible structure of an essentially algebraic theory). I have put notes on this now into the section <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#GeometryAndStructureSheaves">cohesive (oo,1)-topos: Geometry and Structure Sheaves</a>. Nothing deep in that section for the moment, just some simple observations in an attempt to map the territory.

For the time being, I am coming back to seeing how concrete objects may serve to present a canonical geometry (in the sense of: site with compatible structure of an essentially algebraic theory).

I have put notes on this now into the section cohesive (oo,1)-topos: Geometry and Structure Sheaves.

Nothing deep in that section for the moment, just some simple observations in an attempt to map the territory.
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeJun 9th 2011
- (edited Jun 9th 2011)
- PermaLink
Author: Urs
Format: MarkdownItexI have changed the definition of _concrete objects_ in a cohesive $(\infty,1)$-topos. The previous definition demanded that all homotopy sheaves are concrete. But instead it seems that one wants to say that all iterated "atlases" are degreewise concrete. The main statement that 0-truncated objects are concrete precisely if they are concrete sheaves in the ordinary sense remains true. I have put the new definition at [[cohesive (infinity,1)-topos -- structures]] in the section [Concrete objects](http://www.ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#ConcreteObjects). This arose in discussion with [[David Carchedi]], he has done some computations that check that this gives the expected answers for 1-stacks. And it seems clear that his argument generalizes. Eventually I hope to post more details.

I have changed the definition of concrete objects in a cohesive $(\infty,1)$ -topos.

The previous definition demanded that all homotopy sheaves are concrete. But instead it seems that one wants to say that all iterated “atlases” are degreewise concrete.

The main statement that 0-truncated objects are concrete precisely if they are concrete sheaves in the ordinary sense remains true.

I have put the new definition at cohesive (infinity,1)-topos – structures in the section Concrete objects. This arose in discussion with David Carchedi, he has done some computations that check that this gives the expected answers for 1-stacks. And it seems clear that his argument generalizes. Eventually I hope to post more details.
- CommentRowNumber6.
- CommentAuthorzskoda
- CommentTimeJun 9th 2011
- (edited Jun 9th 2011)
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Author: zskoda
Format: MarkdownItex> site with compatible structure of an essentially algebraic theory You often talk about geometric spaces underlying spectra of algebraic theories, but you do not utilize/mention any commutativity condition on the algebraic theory. This is strange, as if you claim that you can do genuine (dream-world) noncommutative geometry.

site with compatible structure of an essentially algebraic theory

You often talk about geometric spaces underlying spectra of algebraic theories, but you do not utilize/mention any commutativity condition on the algebraic theory. This is strange, as if you claim that you can do genuine (dream-world) noncommutative geometry.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeJun 10th 2011
- PermaLink
Author: Urs
Format: MarkdownItexActually, a basic assumption (stated there) of the discussion at [[function algebras on infinity-stacks]] is that the Lawvere theory is abelian in that it receives a morphism from the theory of abelian groups. In that case, the claim is, much of the classical theory goes through.

Actually, a basic assumption (stated there) of the discussion at function algebras on infinity-stacks is that the Lawvere theory is abelian in that it receives a morphism from the theory of abelian groups. In that case, the claim is, much of the classical theory goes through.
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeJun 10th 2011
- (edited Jun 10th 2011)
- PermaLink
Author: Urs
Format: MarkdownItexHm, that's still not the right definition beyond 1-truncated objects. Now I think it should be: $X$ is concrete if either it is 0-truncated and concrete or, recursively, if 1. it has a concrete atlas, meaning an essential epimorphism from a concrete 0-truncated object $Y \to X$; 1. the object $Y \times_X Y$ is concrete.
Hm, that’s still not the right definition beyond 1-truncated objects.

Now I think it should be: $X$ is concrete if either it is 0-truncated and concrete or, recursively, if
1. it has a concrete atlas, meaning an essential epimorphism from a concrete 0-truncated object $Y \to X$ ;
2. the object $Y \times_X Y$ is concrete.
- CommentRowNumber9.
- CommentAuthorzskoda
- CommentTimeJun 10th 2011
- (edited Jun 10th 2011)
- PermaLink
Author: zskoda
Format: MarkdownItexWhat does it mean that the Lawvere theory is abelian (we can not just play words) ? That is commutative in the sense used by Borceux or by Durov ? (I never heard of abelian in this context, like I never heard of an abelian ring). How do you categorify commutativity for the $(n,1)$-theories ? > it receives a morphism from the theory of abelian groups What does that mean for the algebras ? That there is a restriction of scalars functor to the abelian groups ? Like from noncommutative rings to abelian groups, taking the underlying additive group ?

What does it mean that the Lawvere theory is abelian (we can not just play words) ? That is commutative in the sense used by Borceux or by Durov ? (I never heard of abelian in this context, like I never heard of an abelian ring). How do you categorify commutativity for the $(n,1)$ -theories ?

it receives a morphism from the theory of abelian groups

What does that mean for the algebras ? That there is a restriction of scalars functor to the abelian groups ? Like from noncommutative rings to abelian groups, taking the underlying additive group ?
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeJun 10th 2011
- PermaLink
Author: Urs
Format: MarkdownItexSorry, Zoran, I am not playing with words. I think in that entry the precise definition is given. It means that every algebra for the theory has an underlying abelian group. On the other hand, I think you were maybe asking about commutativity. That I did not consider.

Sorry, Zoran, I am not playing with words. I think in that entry the precise definition is given.

It means that every algebra for the theory has an underlying abelian group.

On the other hand, I think you were maybe asking about commutativity. That I did not consider.
- CommentRowNumber11.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
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Author: zskoda
Format: MarkdownItexUrs, a noncommutative ring has an underlying abelian group. How do you create a geometry which is locally one of noncommutative rings ? This sounds like a claim to subsume noncommutative geometry, what is not likely given the naive tools used (which neglect basic inherent problems of noncommutative setup). I mean, I have no problem with the entry on infinity-function theory, but I am suspicious once one starts talking underlying sites and topoi for noncommutative theories. Sites and topoi, are, except in special case, not useful for globalizing noncommutative geometries (unless one forgets the data and takes only the corresponding derived categories of sheaves, what is a loss of info). If it is not just words, then there is an essential answer what is accomplished in noncommutative case, or the extension is just words without genuine examples in mind (experience is that the only examples there fitting are in essence commutative).

Urs, a noncommutative ring has an underlying abelian group. How do you create a geometry which is locally one of noncommutative rings ? This sounds like a claim to subsume noncommutative geometry, what is not likely given the naive tools used (which neglect basic inherent problems of noncommutative setup). I mean, I have no problem with the entry on infinity-function theory, but I am suspicious once one starts talking underlying sites and topoi for noncommutative theories. Sites and topoi, are, except in special case, not useful for globalizing noncommutative geometries (unless one forgets the data and takes only the corresponding derived categories of sheaves, what is a loss of info). If it is not just words, then there is an essential answer what is accomplished in noncommutative case, or the extension is just words without genuine examples in mind (experience is that the only examples there fitting are in essence commutative).
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeJun 12th 2011
- PermaLink
Author: Urs
Format: MarkdownItexI am not sure what you want me to say. The statements at [[function algebras on infinity-stacks]] hold, as stated there, for algebras over Lawvere theories that have underlying abelian groups. That's what I have been saying. I suppose you are after more than that. But I guess I can't say more than that.

I am not sure what you want me to say. The statements at function algebras on infinity-stacks hold, as stated there, for algebras over Lawvere theories that have underlying abelian groups.

That’s what I have been saying. I suppose you are after more than that. But I guess I can’t say more than that.
- CommentRowNumber13.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
- PermaLink
Author: zskoda
Format: MarkdownItexUrs, I was complaining in 6 about talking sites compatible with (essnetially) algebraic theory in noncommutative context. I consider some instances of this artificial. Before we had a discussion when the same thing was stated about generalized schemes a la Lurie where you claimed that one can do the analogues for any algebraic theory. Now you are repeatedly diverting me to (never mentioned by me) infinity function theory. I do not care about infinity function theory and was not ever complaining about it, or its entry (see 6). I care about the repeated claim that the geometry a la Lurie can be used in a straightforward manner with Grothendieck topologies and noncommutative theory. This appeared when we were discussing generalized schemes before, regarding that entry and now when discussing coherent topoi. Putting aside some very special examples (like [[zoranskoda:semicommutative scheme]]), I consider that inclusion of noncommutative case into topos approach to schemes/geometry a misconception, which often propagates in wordings in $n$Lab. (This restriction can possibly be overcome with Q-categories and alike or with loss of information.) You remind me of a talk of Cartier who had a talk at IHES that his approach to functional integration solved the obstacles which were plaguing the subject for years. Todorov was in the audience and after the talk with spectacular abstract complained that the only true problem from the point of view of physicists was with the field theory where there is an infinite dimensional case; and he was mislead by the abstract that there is a breakthrough there.

Urs, I was complaining in 6 about talking sites compatible with (essnetially) algebraic theory in noncommutative context. I consider some instances of this artificial. Before we had a discussion when the same thing was stated about generalized schemes a la Lurie where you claimed that one can do the analogues for any algebraic theory. Now you are repeatedly diverting me to (never mentioned by me) infinity function theory. I do not care about infinity function theory and was not ever complaining about it, or its entry (see 6). I care about the repeated claim that the geometry a la Lurie can be used in a straightforward manner with Grothendieck topologies and noncommutative theory. This appeared when we were discussing generalized schemes before, regarding that entry and now when discussing coherent topoi. Putting aside some very special examples (like semicommutative scheme (zoranskoda)), I consider that inclusion of noncommutative case into topos approach to schemes/geometry a misconception, which often propagates in wordings in $n$ Lab. (This restriction can possibly be overcome with Q-categories and alike or with loss of information.)

You remind me of a talk of Cartier who had a talk at IHES that his approach to functional integration solved the obstacles which were plaguing the subject for years. Todorov was in the audience and after the talk with spectacular abstract complained that the only true problem from the point of view of physicists was with the field theory where there is an infinite dimensional case; and he was mislead by the abstract that there is a breakthrough there.
- CommentRowNumber14.
- CommentAuthorUrs
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
- PermaLink
Author: Urs
Format: MarkdownItexZoran, sorry, I don't know what you are referring to. You keep referring to things that I "often talk about", so I can only guess which entries you mean. It seems to me none of this has to do with the entry that this thread here is about? If you have something concrete that I can focus on, let me know. By the way, I could reverse this kind of argument: you kept and keep saying that non-commutative geometry lies striuctly outside the bounds of standard topos theory, and then instead one needs notions of "Q-sheaves". But recently we saw that these Q-sheaves perfectly and nicely fit into standard topos theory.

Zoran, sorry, I don’t know what you are referring to. You keep referring to things that I “often talk about”, so I can only guess which entries you mean. It seems to me none of this has to do with the entry that this thread here is about?

If you have something concrete that I can focus on, let me know.

By the way, I could reverse this kind of argument: you kept and keep saying that non-commutative geometry lies striuctly outside the bounds of standard topos theory, and then instead one needs notions of “Q-sheaves”. But recently we saw that these Q-sheaves perfectly and nicely fit into standard topos theory.
- CommentRowNumber15.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
- PermaLink
Author: zskoda
Format: MarkdownItexNo, Urs, good question, but it is not true, Q-sheaves do not perfectly fit into standard topos theory (they can be applied in the setup of topoi but also in some other setups, as every needed generalization does), because 1. they are often considered not in topoi but in abelian categories which are not topoi -- they are also important for presheaves which are not sheaves of sets. 2. the inclusion of a Q-category of sheaves into the category of presheaves is NOT left exact. Hence the Q- sheaves of sets **do not** make a topos. 3. the categories of sheaves in a Q-category of presheaves of sets on a category $A$ are not in 1-1 correspondence with Grothendieck topologies in $A$ 4. all the main examples of analogues of Grothendieck topologies on $NAff = (Ass)^{op}$ like nc Zariski, faithfully flat, ffqc, smooth etc. are not Grothendieck topologies but can be formulated in terms of quasitopologies or Q-categories. The main characteristic is that quasipretopoligies and alike formalism allow for the case that it is not a coverage: the pullback of a cover is NOT a cover.
No, Urs, good question, but it is not true, Q-sheaves do not perfectly fit into standard topos theory (they can be applied in the setup of topoi but also in some other setups, as every needed generalization does), because
1. they are often considered not in topoi but in abelian categories which are not topoi – they are also important for presheaves which are not sheaves of sets.
2. the inclusion of a Q-category of sheaves into the category of presheaves is NOT left exact. Hence the Q- sheaves of sets do not make a topos.
3. the categories of sheaves in a Q-category of presheaves of sets on a category $A$ are not in 1-1 correspondence with Grothendieck topologies in $A$
4. all the main examples of analogues of Grothendieck topologies on $NAff = (Ass)^{op}$ like nc Zariski, faithfully flat, ffqc, smooth etc. are not Grothendieck topologies but can be formulated in terms of quasitopologies or Q-categories. The main characteristic is that quasipretopoligies and alike formalism allow for the case that it is not a coverage: the pullback of a cover is NOT a cover.
- CommentRowNumber16.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
- PermaLink
Author: zskoda
Format: MarkdownItex> It seems to me none of this has to do with the entry that this thread here is about? It has, as precisely the entries linked from here talk about structure sheaves and I can not see how this is possible with the generality and tools stated there. I would like to see how it works for any nontrivial noncommutative example. In 6 I said that it looks like you claim a noncommutative geometry is possible in that way, and in 7 you confirm "the Lawvere theory is abelian in that it receives a morphism from the theory of abelian groups. In that case, the claim is, much of the classical theory goes through." Abelian case in your sense includes noncommutative rings, hence that you claim that you can do geometry whose local models are noncommutative rings (or some more complicated abelian Lawvere theories) and where the topology is in the usual Grothendieck sense! I would be happy to see if this were possible in some nontrivial examples (of Grothendieck topologies on NAff, geometries in the sense of Lurie or in Toen's setup). I would like to either see it or having you realize that the lack of examples is a genuine feature and there is a need of generalizations. (I have some ideas, but obviously so far no correspondent here wanting to discuss them.) Edit: recently we had another thread with localizations of algebraic theories and enriched Grothendieck topologies (works of Borceux at el. from 1980s). He has a very clear point after the inital definition where he has to assume that the theory to be commutative in order to have the usual theorems about the role of Lawvere-Tierney or Grothendieck-Gabriel topologies. I hoped they had a bit more (by putting Gabriel's name I hoped they could ocver in some formalism Gabriel localization with sheaf like notions, but alas, not).

It seems to me none of this has to do with the entry that this thread here is about?

It has, as precisely the entries linked from here talk about structure sheaves and I can not see how this is possible with the generality and tools stated there. I would like to see how it works for any nontrivial noncommutative example. In 6 I said that it looks like you claim a noncommutative geometry is possible in that way, and in 7 you confirm “the Lawvere theory is abelian in that it receives a morphism from the theory of abelian groups. In that case, the claim is, much of the classical theory goes through.” Abelian case in your sense includes noncommutative rings, hence that you claim that you can do geometry whose local models are noncommutative rings (or some more complicated abelian Lawvere theories) and where the topology is in the usual Grothendieck sense! I would be happy to see if this were possible in some nontrivial examples (of Grothendieck topologies on NAff, geometries in the sense of Lurie or in Toen’s setup). I would like to either see it or having you realize that the lack of examples is a genuine feature and there is a need of generalizations. (I have some ideas, but obviously so far no correspondent here wanting to discuss them.)

Edit: recently we had another thread with localizations of algebraic theories and enriched Grothendieck topologies (works of Borceux at el. from 1980s). He has a very clear point after the inital definition where he has to assume that the theory to be commutative in order to have the usual theorems about the role of Lawvere-Tierney or Grothendieck-Gabriel topologies. I hoped they had a bit more (by putting Gabriel’s name I hoped they could ocver in some formalism Gabriel localization with sheaf like notions, but alas, not).
- CommentRowNumber17.
- CommentAuthorUrs
- CommentTimeJun 12th 2011
- PermaLink
Author: Urs
Format: MarkdownItex> It has, as precisely the entries linked from here talk about structure sheaves and I can not see how this is possible with the generality and tools stated there. Please give me some more details. If there is a wrong claim somewhere I need to correct it. But I am not sure which claim you say is wrong.

It has, as precisely the entries linked from here talk about structure sheaves and I can not see how this is possible with the generality and tools stated there.

Please give me some more details. If there is a wrong claim somewhere I need to correct it. But I am not sure which claim you say is wrong.
- CommentRowNumber18.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
- PermaLink
Author: zskoda
Format: MarkdownItex> If you have something concrete that I can focus on, let me know. Is asking (as in 16) for a single example of a nontrivial geometry/theory of structured sheaves (1-categorical, preferrably, for stated reasons) based on a site with compatible noncommutative (but abelian in your sense if you wish) theory concrete enough ?

If you have something concrete that I can focus on, let me know.

Is asking (as in 16) for a single example of a nontrivial geometry/theory of structured sheaves (1-categorical, preferrably, for stated reasons) based on a site with compatible noncommutative (but abelian in your sense if you wish) theory concrete enough ?
- CommentRowNumber19.
- CommentAuthorUrs
- CommentTimeJun 12th 2011
- PermaLink
Author: Urs
Format: MarkdownItexThat's concrete enough, yes. Answer: no, I haven't thought much about that. As you know. I am really not sure what you are trying to do with me.

That’s concrete enough, yes. Answer: no, I haven’t thought much about that. As you know.

I am really not sure what you are trying to do with me.
- CommentRowNumber20.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
- PermaLink
Author: zskoda
Format: MarkdownItexMy claim is that no ready examples is not because you did not think enough, but that this is a genuine feature of the subject. Hence stating in all of this algebraic theory without commutative is a somewhat artifical and to some (like me who hope to find the examples of opposite for years) a misleading generality (for local geometry based on Grothendieck topologies). Is it so hard to agree on that and having this in mind ? > I am really not sure what you are trying to do with me. I want here to understand what you are writing in our common sphera of $n$Lab when it concerns my interests. And noncommutative theories are central to my interests as you know. Understanding is not only at the level of wheather a lemma is formally correct but at the level where it is aiming at and what is the power and the set of covered examples of a theory.

My claim is that no ready examples is not because you did not think enough, but that this is a genuine feature of the subject. Hence stating in all of this algebraic theory without commutative is a somewhat artifical and to some (like me who hope to find the examples of opposite for years) a misleading generality (for local geometry based on Grothendieck topologies). Is it so hard to agree on that and having this in mind ?

I am really not sure what you are trying to do with me.

I want here to understand what you are writing in our common sphera of $n$ Lab when it concerns my interests. And noncommutative theories are central to my interests as you know. Understanding is not only at the level of wheather a lemma is formally correct but at the level where it is aiming at and what is the power and the set of covered examples of a theory.
- CommentRowNumber21.
- CommentAuthorUrs
- CommentTimeJun 12th 2011
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Author: Urs
Format: MarkdownItexPlease, give me a spefic point in a specific entry that needs rewriting according to your opinion. Then I can try to react or reply.

Please, give me a spefic point in a specific entry that needs rewriting according to your opinion. Then I can try to react or reply.
- CommentRowNumber22.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
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Author: zskoda
Format: MarkdownItexI am talking in 20 and before about understanding, not rewriting. If I knew how to write your cohesive/infinity-topos circle articles than I would not need help in understanding them (especially their actual, rather than formal, meaning and scope). Of course, I do have good idea how to provide some noncommutative examples in slightly extended formalism, but this would clearly be burden to you in current direction of work. I did not intend this: I hoped you do have some intermediate case/example (like my ideas of [[zoranskoda:semicommutative scheme]] which belong to nc world but I believe give examples of your setup (and many other slight modifications of commutative)).

I am talking in 20 and before about understanding, not rewriting. If I knew how to write your cohesive/infinity-topos circle articles than I would not need help in understanding them (especially their actual, rather than formal, meaning and scope).

Of course, I do have good idea how to provide some noncommutative examples in slightly extended formalism, but this would clearly be burden to you in current direction of work. I did not intend this: I hoped you do have some intermediate case/example (like my ideas of semicommutative scheme (zoranskoda) which belong to nc world but I believe give examples of your setup (and many other slight modifications of commutative)).
- CommentRowNumber23.
- CommentAuthorUrs
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
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Author: Urs
Format: MarkdownItexAh, so you want examples of cohesive toposes for noncommutative geometries? It seemed to me that using that Q-sheaves can be thought of as objects making one of the two natural transformations in a cohesive topos become an equivalence, one gets a nice characterization of these as objects in a cohesive presheaf topos. I think with what we worked out at [[Q-category]] that's pretty immediate. But I don't have time to expand on this right now. On something different but related: I started wondering if the wide-spread idea that quantization is about non-commutative geometry is actually right. If one follows Bates-Weinstein's _Lectures on symplectic geometry_ , page 80, then we are to read a quantum $\ast$-algebra as being a non-associative Poisson algebra with symmetric but non-associative Jordan product $(a,b) = a b + b a$ and Lie bracket $[a, b] = a b - b a$. Of course by itself that's just a game with symbols. But it indicates that the idea that quantization is about making phase space non-commutative is not necessarily right.

Ah, so you want examples of cohesive toposes for noncommutative geometries? It seemed to me that using that Q-sheaves can be thought of as objects making one of the two natural transformations in a cohesive topos become an equivalence, one gets a nice characterization of these as objects in a cohesive presheaf topos. I think with what we worked out at Q-category that’s pretty immediate. But I don’t have time to expand on this right now.

On something different but related: I started wondering if the wide-spread idea that quantization is about non-commutative geometry is actually right. If one follows Bates-Weinstein’s Lectures on symplectic geometry , page 80, then we are to read a quantum $\ast$ -algebra as being a non-associative Poisson algebra with symmetric but non-associative Jordan product $(a,b) = a b + b a$ and Lie bracket $[a, b] = a b - b a$ .

Of course by itself that’s just a game with symbols. But it indicates that the idea that quantization is about making phase space non-commutative is not necessarily right.
- CommentRowNumber24.
- CommentAuthorzskoda
- CommentTimeJun 12th 2011
- (edited Jun 12th 2011)
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Author: zskoda
Format: MarkdownItexThis about Bates-Weinstein is an interesting remark. I am not personally fond of the idea that quantum= noncommutative; I think these are quite different axes. I understand that the *presheaf* categories involved in Q-categories defining the category of sheaves of sets on a Q-category make a cohesive topos under mild completness assumptions on the underlying category, but the Q-sheaf category itself is *not* a topos and it seems to me that little is known on Q-sheaves when the infinity Q-category is not induced from a Grothendieck topology. I am looking either for an example of a nontrivial ordinary (Grothendieck) site of noncommutative spaces (which you seem not to have), where your formalism could directly apply, or the (what I still hope) a modification of your formalism where the nc spaces would be sheaves on a Q-category which are locally representable. I mean, Lurie has all that pregeometry, geometry, admissibility business etc. I think if one modifies all that for having in mind sheaves on a Q-category than one might have lots of applications to noncommutative Lawvere theories. Lurie's formalism suggests that it is natural to look for an underlying higher topos in whatever case of a generalized "scheme". Now it would be extremely interesting if one can see something like that for noncommutative examples. As far as more research level digression: I also have a genuine (and important) example of a noncommutative space defined by means of what is eventually stable category of qcoh sheaves, which is glued well in certain Bousfield sense out of categories of qcoh sheaves on affine spaces: this affine spaces make a cover in noncommutative derived sense. My problem is in the combination (I know how to deal with nc covers but not derived case of it). Now I would like that this space is defined as a simplicial Q-sheaf on NAff which is locally representable. This is not easy, but one of the things which should not be that difficult is that one at least knows what is the natural candidate for the category of Q-sheaves on NAff which are locally representable (locally in the sense of nc covers defined in terms of inducing a conservative family of Bousfield localizations of the stable category of qcoh sheaves).

This about Bates-Weinstein is an interesting remark. I am not personally fond of the idea that quantum= noncommutative; I think these are quite different axes.

I understand that the presheaf categories involved in Q-categories defining the category of sheaves of sets on a Q-category make a cohesive topos under mild completness assumptions on the underlying category, but the Q-sheaf category itself is not a topos and it seems to me that little is known on Q-sheaves when the infinity Q-category is not induced from a Grothendieck topology.

I am looking either for an example of a nontrivial ordinary (Grothendieck) site of noncommutative spaces (which you seem not to have), where your formalism could directly apply, or the (what I still hope) a modification of your formalism where the nc spaces would be sheaves on a Q-category which are locally representable. I mean, Lurie has all that pregeometry, geometry, admissibility business etc. I think if one modifies all that for having in mind sheaves on a Q-category than one might have lots of applications to noncommutative Lawvere theories.

Lurie’s formalism suggests that it is natural to look for an underlying higher topos in whatever case of a generalized “scheme”. Now it would be extremely interesting if one can see something like that for noncommutative examples.

As far as more research level digression:

I also have a genuine (and important) example of a noncommutative space defined by means of what is eventually stable category of qcoh sheaves, which is glued well in certain Bousfield sense out of categories of qcoh sheaves on affine spaces: this affine spaces make a cover in noncommutative derived sense. My problem is in the combination (I know how to deal with nc covers but not derived case of it). Now I would like that this space is defined as a simplicial Q-sheaf on NAff which is locally representable. This is not easy, but one of the things which should not be that difficult is that one at least knows what is the natural candidate for the category of Q-sheaves on NAff which are locally representable (locally in the sense of nc covers defined in terms of inducing a conservative family of Bousfield localizations of the stable category of qcoh sheaves).
- CommentRowNumber25.
- CommentAuthorzskoda
- CommentTimeJun 13th 2011
- (edited Jun 13th 2011)
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Author: zskoda
Format: MarkdownItexShort question to Urs: If you have an abelian Lawvere theory, and consider the category of modules over a fixed algebra of that theory (that is the category of qcoh sheaves in affine case), is that category automatically abelian ? (I think I have seen that somewhere in $n$Lab but can not find). Comment: For commutative theories in the sense of Durov it is possible to do geometric spectra and to do gluing via usual Grothendieck sites, unlike for noncommutative theories, but on the other hand the qcoh sheaves do not necessarily form abelian categories. To define the category of qcoh sheaves, one takes Zariski topology defined in terms of flat reflective localizations (with some finiteness condition, which I did not see in Durov, I think, but is explained in Rosenberg somewhere).

Short question to Urs: If you have an abelian Lawvere theory, and consider the category of modules over a fixed algebra of that theory (that is the category of qcoh sheaves in affine case), is that category automatically abelian ? (I think I have seen that somewhere in $n$ Lab but can not find).

Comment: For commutative theories in the sense of Durov it is possible to do geometric spectra and to do gluing via usual Grothendieck sites, unlike for noncommutative theories, but on the other hand the qcoh sheaves do not necessarily form abelian categories. To define the category of qcoh sheaves, one takes Zariski topology defined in terms of flat reflective localizations (with some finiteness condition, which I did not see in Durov, I think, but is explained in Rosenberg somewhere).

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