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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexrecently Zoran and I have been discussing how Rosenberg and Kontsevich have formalized [[cohesive (infinity,1)-topos -- infinitesimal cohesion|infinitesimal cohesion]] in terms of systems of adjunctions that they called [[Q-categories]] (see there for details). We haven't yet discussed the following point: a little reflection shows that the basic mechanism of the Rosenberg-Kontsevich notion of infintesimal thickening is the same as that in Lurie's [[Deformation Theory]]: RK consider for some category $\mathcal{C}$ of algebras first the domain/codomain fibration $$ \mathcal{C} \stackrel{\overset{cod}{\leftarrow}}{\stackrel{\overset{i}{\to}}{\underset{dom}{\leftarrow}}} Func(\Delta[1], \mathcal{C}) $$ (where the lower adjoint pair exhibits a Q-category, since the diagonal $i$ is full and faithful) and then pick inside the arrow category on the right a subcategory of those morphisms which may be thought of as exhibiting their domain as a dual infinitesimal thickening of their codomain. Now, the basic idea of Lurie's [[Deformation Theory]] is that, following Quillen's old insight that [[Mod]] is a [[tangent category]], there is a canonical way to pick such a "subcategory": take the category $T_{\mathcal{C}}$ of fiberwise abelian group objects aka stable objects. It's not exactly a subcategory, but it does have a forgetful functor $T_{\mathcal{C}} \to Func(\Delta[1], \mathcal{C})$. I had played around with trying to incorpirate [[Deformation Theory]] into the context of [[cohesive topos]]es before, but in view of the above I'll try to have a fresh look at it again. This suggests that [[cohesive (infinity,1)-topos -- infinitesimal cohesion|infinitesimal cohesion]] is not necessarily extra structure after all, but that there is a canonical infinitesimal cohesive neighbourhood for every cohesive $\infty$-topos.

   recently Zoran and I have been discussing how Rosenberg and Kontsevich have formalized infinitesimal cohesion in terms of systems of adjunctions that they called Q-categories (see there for details).

   We haven’t yet discussed the following point: a little reflection shows that the basic mechanism of the Rosenberg-Kontsevich notion of infintesimal thickening is the same as that in Lurie’s Deformation Theory:

   RK consider for some category $\mathcal{C}$ of algebras first the domain/codomain fibration

   $$\mathcal{C} \stackrel{\overset{cod}{\leftarrow}}{\stackrel{\overset{i}{\to}}{\underset{dom}{\leftarrow}}} Func(\Delta[1], \mathcal{C})$$

   (where the lower adjoint pair exhibits a Q-category, since the diagonal $i$ is full and faithful) and then pick inside the arrow category on the right a subcategory of those morphisms which may be thought of as exhibiting their domain as a dual infinitesimal thickening of their codomain.

   Now, the basic idea of Lurie’s Deformation Theory is that, following Quillen’s old insight that Mod is a tangent category, there is a canonical way to pick such a “subcategory”: take the category $T_{\mathcal{C}}$ of fiberwise abelian group objects aka stable objects. It’s not exactly a subcategory, but it does have a forgetful functor $T_{\mathcal{C}} \to Func(\Delta[1], \mathcal{C})$.

   I had played around with trying to incorpirate Deformation Theory into the context of cohesive toposes before, but in view of the above I’ll try to have a fresh look at it again. This suggests that infinitesimal cohesion is not necessarily extra structure after all, but that there is a canonical infinitesimal cohesive neighbourhood for every cohesive $\infty$-topos.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2011
   • (edited May 3rd 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe to make this work out as it looks it should, we need to restrict to "connective tangent $\infty$-categories". In the following sense: By the discussion in section 1.1 of _[[Deformation Theory]]_ the [[tangent (infinity,1)-category]] of $\mathcal{C}$ fits into the diagram displayed above as $$ \mathcal{C} \stackrel{\overset{cod}{\leftarrow}}{\stackrel{\overset{i}{\to}}{\underset{dom}{\leftarrow}}} Func(\Delta[1], \mathcal{C}) \stackrel{\overset{L}{\to}}{\underset{u}{\leftarrow}} T_{\mathcal{C}} \,, $$ where the pair of functors on the right forms not a plain adjunction, but a $cod$-relative adjunction (section 1.2). The composite $L \circ i : \mathcal{C} \to T_{\mathcal{C}}$ has the interpretation of producing the cotangent complex (derived Kähler differentials) of an algebra $A \in \mathcal{C}$. The functor $u$ on the othe hand is (by example 1.1.4) fiberwise the $\Omega^\infty$-functor that sends [[spectrum object]]s in $Func(\Delta[1], \mathcal{C})_{f}$ to their underlying object in degree 0 . (The corresponding "connective spectrum"). Since for the purposes of the setup of [[infinitesimal cohesion]] I seem to need the $(cod \dashv i)$-adjunction, and since I don't see how or even that this would prolong to $T_{\mathcal{C}}$, I am wondering if a sensible idea would be for that purpose to just restrict the $(cod \dashv i)$-adjunction to the image of $u : T_{\mathcal{C}} \to Func(\Delta[1], \mathcal{C})$, hence in words to the subcategory of $Func(\Delta[1], \mathcal{C})$ on the objects that exhibit their domain as connective $\infty$-abelian groups over their codomain. (In the underived case, where $\mathcal{C}$ is a 1-category, the difference would disappear anyway.)

   Maybe to make this work out as it looks it should, we need to restrict to "connective tangent $\infty$-categories". In the following sense:

   By the discussion in section 1.1 of Deformation Theory the tangent (infinity,1)-category of $\mathcal{C}$ fits into the diagram displayed above as

   $$\mathcal{C} \stackrel{\overset{cod}{\leftarrow}}{\stackrel{\overset{i}{\to}}{\underset{dom}{\leftarrow}}} Func(\Delta[1], \mathcal{C}) \stackrel{\overset{L}{\to}}{\underset{u}{\leftarrow}} T_{\mathcal{C}} \,,$$

   where the pair of functors on the right forms not a plain adjunction, but a $cod$-relative adjunction (section 1.2). The composite $L \circ i : \mathcal{C} \to T_{\mathcal{C}}$ has the interpretation of producing the cotangent complex (derived Kähler differentials) of an algebra $A \in \mathcal{C}$.

   The functor $u$ on the othe hand is (by example 1.1.4) fiberwise the $\Omega^\infty$-functor that sends spectrum objects in $Func(\Delta[1], \mathcal{C})_{f}$ to their underlying object in degree 0 . (The corresponding "connective spectrum").

   Since for the purposes of the setup of infinitesimal cohesion I seem to need the $(cod \dashv i)$-adjunction, and since I don’t see how or even that this would prolong to $T_{\mathcal{C}}$, I am wondering if a sensible idea would be for that purpose to just restrict the $(cod \dashv i)$-adjunction to the image of $u : T_{\mathcal{C}} \to Func(\Delta[1], \mathcal{C})$, hence in words to the subcategory of $Func(\Delta[1], \mathcal{C})$ on the objects that exhibit their domain as connective $\infty$-abelian groups over their codomain.

   (In the underived case, where $\mathcal{C}$ is a 1-category, the difference would disappear anyway.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, theorem 1.5.19 should help to give the prolonged adjunction $(cod \ashv i) : \mathcal{C} \to T_{\mathcal{C}}$ in the case that $\mathcal{C}$ is the category of algebras over a [[coherent (infinity,1)-operad]]. See [here](http://nlab.mathforge.org/nlab/show/tangent+%28infinity%2C1%29-category#RelationToModules).

   Ah, theorem 1.5.19 should help to give the prolonged adjunction $(cod \ashv i) : \mathcal{C} \to T_{\mathcal{C}}$ in the case that $\mathcal{C}$ is the category of algebras over a coherent (infinity,1)-operad.

   See here.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 3rd 2011
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   Author: zskoda
   Format: MarkdownItexI am not quite understanding what you are doing above. I should just mention that I recall that you were using tangeng category in the above stable sense also for getting the category of qcoh sheaves. But for modules over generalized rings (that is a finitary monad in sets) one does not get an abelian category of qcoh sheaves. So for a general operad I would not expect that the category of qcoh sheaves is stable. On the other hand, for some other purposes the stabilization may be good even in that generality. I do not know how you address this.

   I am not quite understanding what you are doing above. I should just mention that I recall that you were using tangeng category in the above stable sense also for getting the category of qcoh sheaves. But for modules over generalized rings (that is a finitary monad in sets) one does not get an abelian category of qcoh sheaves. So for a general operad I would not expect that the category of qcoh sheaves is stable. On the other hand, for some other purposes the stabilization may be good even in that generality. I do not know how you address this.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I am not quite understanding what you are doing above. Okay, so let's see where I am getting incomprehensible. The first statement is that both in Rosenberg-Kontsevich and in Quillen-Lurie we identify infinitesimal thickenings of duals of algebras as a subcategory of the arrow category of the category of algebras. The second statement is that there is a _canonical_ such subcategory: the image of the tangent category. And, yes, the relation to quasicoherent $\infty$-stacks I am hoping to exploit in this. But let's see.

   I am not quite understanding what you are doing above.

   Okay, so let’s see where I am getting incomprehensible.

   The first statement is that both in Rosenberg-Kontsevich and in Quillen-Lurie we identify infinitesimal thickenings of duals of algebras as a subcategory of the arrow category of the category of algebras.

   The second statement is that there is a canonical such subcategory: the image of the tangent category.

   And, yes, the relation to quasicoherent $\infty$-stacks I am hoping to exploit in this. But let’s see.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeMay 4th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> The first statement is that both in Rosenberg-Kontsevich and in Quillen-Lurie we identify infinitesimal thickenings of duals of algebras as a subcategory of the arrow category of the category of algebras. It is a long way before I really understand the story, but let us start here. The case of ordinary algebras $Alg_k$ is rather classical. Now both Lurie and KR talk these notions in bigger generality. Now, for which generality you claim there is an agreement ? This is also related to Quillen insight on tangent category. I do not expect at first that precisely that recipe should hold for any operad, even without a commutativity assumption.

   The first statement is that both in Rosenberg-Kontsevich and in Quillen-Lurie we identify infinitesimal thickenings of duals of algebras as a subcategory of the arrow category of the category of algebras.

   It is a long way before I really understand the story, but let us start here. The case of ordinary algebras $Alg_k$ is rather classical. Now both Lurie and KR talk these notions in bigger generality. Now, for which generality you claim there is an agreement ? This is also related to Quillen insight on tangent category. I do not expect at first that precisely that recipe should hold for any operad, even without a commutativity assumption.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 4th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Now both Lurie and KR talk these notions in bigger generality. Now, for which generality you claim there is an agreement ? The tangent catgeory of $Alg_k$ will not agree with the category of morphisms with nilpotent kernel: the tangent category will give suare-0-extensions only hence only first order nilpotent kernels. But apart from this the crucial agreement is in the general structure of the theory: in both cases the adjoint triple into the arrow category turns into an adjoint pair/triple that exhibits infinitesimal extension, and the adjunctions carry all the crucial information.

   Now both Lurie and KR talk these notions in bigger generality. Now, for which generality you claim there is an agreement ?

   The tangent catgeory of $Alg_k$ will not agree with the category of morphisms with nilpotent kernel: the tangent category will give suare-0-extensions only hence only first order nilpotent kernels.

   But apart from this the crucial agreement is in the general structure of the theory: in both cases the adjoint triple into the arrow category turns into an adjoint pair/triple that exhibits infinitesimal extension, and the adjunctions carry all the crucial information.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeMay 5th 2011
   • (edited May 5th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThere is sometimes a way to iterate the first order neighborhoods to higher ones. In the language of abelian categories one uses the [[Gabriel multiplication]] of [[topologizing subcategories]] (edit: link corrected) for that.

   There is sometimes a way to iterate the first order neighborhoods to higher ones. In the language of abelian categories one uses the Gabriel multiplication of topologizing subcategories (edit: link corrected) for that.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMay 5th 2011
   • (edited May 5th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> There is sometimes a way to iterate the first order neighborhoods to higher ones I was thinking about that, too. This can be useful. For instant the tangent Lie algebroid $T X$, is the de Rham space of $X$ precisely if our notion of inifnitesimal cohesion involves only first order infinitesimals. But then one should iterate: $T X$ is dual to forms, its iterations will be dual to "differential gorms" and eventually "differential worms" as [here](http://arxiv.org/abs/math/0307303)

   There is sometimes a way to iterate the first order neighborhoods to higher ones

   I was thinking about that, too. This can be useful.

   For instant the tangent Lie algebroid $T X$, is the de Rham space of $X$ precisely if our notion of inifnitesimal cohesion involves only first order infinitesimals. But then one should iterate: $T X$ is dual to forms, its iterations will be dual to “differential gorms” and eventually “differential worms” as here

10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeMay 5th 2011
    • (edited May 5th 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexIn Abelian context (i.e. space represented by their abelian categories of qcoh sheaves), the good feature for various structures is that [[Gabriel multiplication]] on the class additive subcategories preserves the classes of [[topologizing subcategories]], of [[reflective subcategory|reflective]] topologizing subcategories, and of coreflective toplogizing subcategories. This makes iteration with powers by Gabriel multiplication compatible with various geometric notions.

    In Abelian context (i.e. space represented by their abelian categories of qcoh sheaves), the good feature for various structures is that Gabriel multiplication on the class additive subcategories preserves the classes of topologizing subcategories, of reflective topologizing subcategories, and of coreflective toplogizing subcategories. This makes iteration with powers by Gabriel multiplication compatible with various geometric notions.

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeMay 5th 2011
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    Author: zskoda
    Format: MarkdownItexNew entry [[neighborhood of a topologizing subcategory]].

    New entry neighborhood of a topologizing subcategory.