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1. • CommentRowNumber1.
   • CommentAuthorMichael_Bachtold
   • CommentTimeApr 23rd 2010
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   Author: Michael_Bachtold
   Format: TextDear nLab architects, the n in my nPOV is still 1, but I'm trying to raise that number by reading the Lab lately. In particular, as I struggle to understand what "derived mathematics" is conceptually all about, I've been wanting to formulate a question. Since I feel it unsuited for MO and would like to eventually find the answer in the nLab I hope it's OK for me to come here and ask it. The rough picture I have, of what people do when they pass from a usual category C (say abelian, or a non-linear category like schemes) to its derived category, is that they embed C into an (infinity,1)-category D(C) in such a way that no information is lost and one may even see finer details. Now I've seen plenty of evidence to believe in the usefulness and necessity of this construction, hence I'm motivated enough to learn and understand it. On the other hand I can't see a pattern in the problems it solves, which makes it feel somewhat Ad hoc. A similar feeling I get looking at the detailed construction of such derived categories. Some have the form: embed the classical category C into the category of simplicial objects in C an transform this into an infinity category. For someone like me who is not very intimate with simplicial objects, this raises the simple minded question: why could someone else not come up with another way to transform a category into an (infty,1)-category which solves similar (or other) nice problems? All this may allow a precise answer if asked as follows: Does the (infinity,1)-functor which embeds the classical 1-category (abelian or of schemes etc.) into its derived category satisfy a universal property? (maybe I'm erroneously assuming there's just one way of constructing a derived category of some category. A least for abelian categories this seems to be so, how is it derived algebraic geometry?) Ideally I expect two things from such a universal property: 1) that it allows one to deduce by (higher) abstract nonsense all the other reasons why people like derived stuff so much. 2) that it even explains why one is forced to leave the dear world of 1-categories to the world of higher categories. Allow me to go on and tell you what kind of answer I would love to hear. For classical derived categories from Homological Algebra I've seen the following motivation: several functors (like Hom and tensor) are not exact when applied to the original category, but passing to the derived category they become somehow exact. This suggest that derived categories might be the universal solution to making functors commute with limits and colimits. I don't attempt to make this precise since I lack the background of higher categories, and maybe someone here can tell me immediately that this is wrong. Anyway as we are not in MO I take the liberty to add some follow-up questions: 3) What type of categories can one actually derive? Certainly abelian ones, and then there are categories of schemes and manifolds. Are these toposes? 4) Can one recover the original 1-category from its derived category? And how is this done? I know that in the classical theory of derived categories one needs a t-structure. What additional structure on the derived category does one need in the (infty,1) setting? I suspect that some of these questions are addressed in Luries work, for example he gives a universal property of the derived category of an abelian category, but I'm not able to digest what it's saying. So any answers which could partially enlighten me (or correct my wrong worldview) are much appreciated. P.S. while I wrote this question I found an article of Keller from 91 "Derived categories and universal problems" addressing the same question for the classical setting of derived categories. By reading the introduction he seems to conclude that derived categories alone don't give an answer. Which is not surprising after all I've heard about triangulated categories just being a shadow of the true infinity derived version.

   Dear nLab architects,
   
   the n in my nPOV is still 1, but I'm trying to raise that number by reading the Lab lately. In particular, as I struggle to understand what "derived mathematics" is conceptually all about, I've been wanting to formulate a question. Since I feel it unsuited for MO and would like to eventually find the answer in the nLab I hope it's OK for me to come here and ask it.
   
   The rough picture I have, of what people do when they pass from a usual category C (say abelian, or a non-linear category like schemes) to its derived category, is that they embed C into an (infinity,1)-category D(C) in such a way that no information is lost and one may even see finer details.
   
   Now I've seen plenty of evidence to believe in the usefulness and necessity of this construction, hence I'm motivated enough to learn and understand it. On the other hand I can't see a pattern in the problems it solves, which makes it feel somewhat Ad hoc. A similar feeling I get looking at the detailed construction of such derived categories. Some have the form: embed the classical category C into the category of simplicial objects in C an transform this into an infinity category. For someone like me who is not very intimate with simplicial objects, this raises the simple minded question: why could someone else not come up with another way to transform a category into an (infty,1)-category which solves similar (or other) nice problems?
   
   All this may allow a precise answer if asked as follows:
   
   Does the (infinity,1)-functor which embeds the classical 1-category (abelian or of schemes etc.) into its derived category satisfy a universal property?
   
   (maybe I'm erroneously assuming there's just one way of constructing a derived category of some category. A least for abelian categories this seems to be so, how is it derived algebraic geometry?)
   
   Ideally I expect two things from such a universal property:
   
   1) that it allows one to deduce by (higher) abstract nonsense all the other reasons why people like derived stuff so much.
   
   2) that it even explains why one is forced to leave the dear world of 1-categories to the world of higher categories.
   
   Allow me to go on and tell you what kind of answer I would love to hear.
   For classical derived categories from Homological Algebra I've seen the following motivation: several functors (like Hom and tensor) are not exact when applied to the original category, but passing to the derived category they become somehow exact. This suggest that derived categories might be the universal solution to making functors commute with limits and colimits. I don't attempt to make this precise since I lack the background of higher categories, and maybe someone here can tell me immediately that this is wrong.
   
   Anyway as we are not in MO I take the liberty to add some follow-up questions:
   
   3) What type of categories can one actually derive? Certainly abelian ones, and then there are categories of schemes and manifolds. Are these toposes?
   
   4) Can one recover the original 1-category from its derived category? And how is this done? I know that in the classical theory of derived categories one needs a t-structure. What additional structure on the derived category does one need in the (infty,1) setting?
   
   I suspect that some of these questions are addressed in Luries work, for example he gives a universal property of the derived category of an abelian category, but I'm not able to digest what it's saying.
   
   So any answers which could partially enlighten me (or correct my wrong worldview) are much appreciated.
   
   P.S. while I wrote this question I found an article of Keller from 91 "Derived categories and universal problems" addressing the same question for the classical setting of derived categories. By reading the introduction he seems to conclude that derived categories alone don't give an answer. Which is not surprising after all I've heard about triangulated categories just being a shadow of the true infinity derived version.
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 23rd 2010
   • (edited Apr 23rd 2010)
   • PermaLink
   Author: Urs
   Format: Markdown> embed the classical category C into the category of simplicial objects in C an transform this into an infinity category. For someone like me who is not very intimate with simplicial objects, this raises the simple minded question: why could someone else not come up with another way to transform a category into an (infty,1)-category which solves similar (or other) nice problems? A quick answer is: passing to simplicial objects in a category corresponds to passing to the (oo,1)-category of [[internal infinity-groupoid]]s in that category, as also described at [[Dold-Kan correspondence]]. That is evidently a natural oo-thing to do. I'll try to say more later. Have no time now...

   embed the classical category C into the category of simplicial objects in C an transform this into an infinity category. For someone like me who is not very intimate with simplicial objects, this raises the simple minded question: why could someone else not come up with another way to transform a category into an (infty,1)-category which solves similar (or other) nice problems?

   A quick answer is: passing to simplicial objects in a category corresponds to passing to the (oo,1)-category of internal infinity-groupoids in that category, as also described at Dold-Kan correspondence. That is evidently a natural oo-thing to do.

   I'll try to say more later. Have no time now...

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeApr 23rd 2010
   • (edited Apr 23rd 2010)
   • PermaLink
   Author: zskoda
   Format: Markdown> Can one recover the original 1-category from its derived category? And how is this done? No. You would need an additional structure, called t-structure. Infinity version is sketched in Lurie's stable infinity categories paper. > Certainly abelian ones, and then there are categories of schemes and manifolds. Are these toposes? Most abelian categories of modules and of quasicoherent sheaves are not topoi. Schemes can be in functor of points approach considered as set-valued functors so can be embedded into categories of sheaves on Aff, which are topoi.

   Can one recover the original 1-category from its derived category? And how is this done?

   No. You would need an additional structure, called t-structure. Infinity version is sketched in Lurie's stable infinity categories paper.

   Certainly abelian ones, and then there are categories of schemes and manifolds. Are these toposes?

   Most abelian categories of modules and of quasicoherent sheaves are not topoi. Schemes can be in functor of points approach considered as set-valued functors so can be embedded into categories of sheaves on Aff, which are topoi.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeApr 23rd 2010
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   Author: Mike Shulman
   Format: MarkdownItexWelcome, Michael! You've asked a good question, but nevertheless I think it's not quite the right question. Let's start with a different, simpler, question: when do we need to use category theory? One good answer is: when we want to study objects that naturally form a category. The generalization to higher category theory is then immediate: when do we need to use $(\infty,1)$-categories? When we want to study objects that naturally form an $(\infty,1)$-category. From this point of view, the process of "deriving" can be described as follows: we've been studying some objects that form a category, but we notice that some constructions we want to perform on those objects produce a new kind of object, and these new kind of objects naturally form an $(\infty,1)$-category. For instance, we may want to take the quotient of a scheme by the action of a group --- but that quotient may not exist as another scheme, and when it does it may not be what we would want it to be (it loses too much information). Thus we need to consider some new kind of object, called "stacks," in which we can form a "good" quotient. But stacks don't form a 1-category, only an $(\infty,1)$-category (or at least a (2,1)-category, depending on what kind of stack you want). Or perhaps we want to study some functor $F:A\to B$ between abelian categories, but we notice that $F$ isn't really as well-behaved as we'd like it to be, e.g. it's not exact. Somehow what we really want to study is not the functor $F$ itself, but some similar, better-behaved functor that "looks like $F$." We know at least that when applied to well-behaved objects (e.g. projective, or injective), $F$ is well-behaved, so we wonder whether we can replace any object by one that is well-behaved. Maybe we can cover any object by a projective one, but the object contains more information than its projective cover, namely the relations we have to quotient by. And those relations should then be presented by a projective cover, which in turn has relations between relations, and so on. Thus we are led to replace the object we started with by a projective resolution of it, and $F$ by a related "derived" functor which acts on projective-resolution-like-objects, i.e. chain complexes. But these naturally form an $(\infty,1)$-category, not just a category, and so the derived functor of $F$ must also be an $(\infty,1)$-functor. There's nothing in this that says that there's only one way to "$(\infty,1)$-ize" a given category, and I wouldn't expect there to be, although some specific constructions have specific universal properties. But for instance, a given abelian category actually has several derived categories, such as bounded above, bounded below, unbounded. And in fact even if you fix the unbounded case, there are different choices you could make: you can invert all quasi-isomorphisms, or some smaller class of maps. The former at least used to be the most common, but in the case of abelian sheaves, I believe the former corresponds to "hypercompletion," so that in many cases it may be better to do the latter instead. That said, certain constructions do come up again and again, like simplicial objects. One way to think about that is that it's like the Yoneda lemma. If you want to embed a category in a better-behaved one, one good way to do it is with its Yoneda embedding into its category of presheaves. (Or maybe sheaves, if your category has some more structure.) In particular, if you want to take colimits that don't exist in your category, this is a good way to do it, because the Yoneda embedding is the free cocompletion. Likewise, any 1-category can be considered as an $(\infty,1)$-category, but it won't be a very well-behaved one; but you can embed it in its category of $(\infty,1)$-(pre)sheaves which will be better-behaved. In particular, if you want to take colimits, like group action quotients as I mentioned above, then this is a good thing to do. Simplicial objects in a category are just a particular "representable" class of $(\infty,1)$-(pre)sheaves. Does that help?

   Welcome, Michael!

   You’ve asked a good question, but nevertheless I think it’s not quite the right question. Let’s start with a different, simpler, question: when do we need to use category theory? One good answer is: when we want to study objects that naturally form a category. The generalization to higher category theory is then immediate: when do we need to use $(\infty,1)$-categories? When we want to study objects that naturally form an $(\infty,1)$-category.

   From this point of view, the process of “deriving” can be described as follows: we’ve been studying some objects that form a category, but we notice that some constructions we want to perform on those objects produce a new kind of object, and these new kind of objects naturally form an $(\infty,1)$-category.

   For instance, we may want to take the quotient of a scheme by the action of a group — but that quotient may not exist as another scheme, and when it does it may not be what we would want it to be (it loses too much information). Thus we need to consider some new kind of object, called “stacks,” in which we can form a “good” quotient. But stacks don’t form a 1-category, only an $(\infty,1)$-category (or at least a (2,1)-category, depending on what kind of stack you want).

   Or perhaps we want to study some functor $F:A\to B$ between abelian categories, but we notice that $F$ isn’t really as well-behaved as we’d like it to be, e.g. it’s not exact. Somehow what we really want to study is not the functor $F$ itself, but some similar, better-behaved functor that “looks like $F$.” We know at least that when applied to well-behaved objects (e.g. projective, or injective), $F$ is well-behaved, so we wonder whether we can replace any object by one that is well-behaved. Maybe we can cover any object by a projective one, but the object contains more information than its projective cover, namely the relations we have to quotient by. And those relations should then be presented by a projective cover, which in turn has relations between relations, and so on. Thus we are led to replace the object we started with by a projective resolution of it, and $F$ by a related “derived” functor which acts on projective-resolution-like-objects, i.e. chain complexes. But these naturally form an $(\infty,1)$-category, not just a category, and so the derived functor of $F$ must also be an $(\infty,1)$-functor.

   There’s nothing in this that says that there’s only one way to “$(\infty,1)$-ize” a given category, and I wouldn’t expect there to be, although some specific constructions have specific universal properties. But for instance, a given abelian category actually has several derived categories, such as bounded above, bounded below, unbounded. And in fact even if you fix the unbounded case, there are different choices you could make: you can invert all quasi-isomorphisms, or some smaller class of maps. The former at least used to be the most common, but in the case of abelian sheaves, I believe the former corresponds to “hypercompletion,” so that in many cases it may be better to do the latter instead.

   That said, certain constructions do come up again and again, like simplicial objects. One way to think about that is that it’s like the Yoneda lemma. If you want to embed a category in a better-behaved one, one good way to do it is with its Yoneda embedding into its category of presheaves. (Or maybe sheaves, if your category has some more structure.) In particular, if you want to take colimits that don’t exist in your category, this is a good way to do it, because the Yoneda embedding is the free cocompletion. Likewise, any 1-category can be considered as an $(\infty,1)$-category, but it won’t be a very well-behaved one; but you can embed it in its category of $(\infty,1)$-(pre)sheaves which will be better-behaved. In particular, if you want to take colimits, like group action quotients as I mentioned above, then this is a good thing to do. Simplicial objects in a category are just a particular “representable” class of $(\infty,1)$-(pre)sheaves.

   Does that help?

5. • CommentRowNumber5.
   • CommentAuthorMichael_Bachtold
   • CommentTimeApr 26th 2010
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   Author: Michael_Bachtold
   Format: TextHi Urs, Zoran and Mike. Thanks for your answers! > Does that help? Quite a bit I should hope. So in particular from Mikes answer I get the feeling that "deriving" is not so much about seeing finer details, rather its about correcting defects the "traditional" categories had by replacing them with better behaved ones. (Let's for the sake of this discussion assume that all 1-categories are understood as $(\infty,1)$-categories in the natural way (i.e. lets say I've already been converted to the higher categorical point of view :-)). If from Zorans answer I may conclude that one cannot recover the traditional category from a derived version of it, would it then be correct (though somewhat radical) to say the following: the traditional categories people have been looking at (e.g. of schemes, of abelian sheaves etc) were wrong all along (mainly due to the fact that the higher categorical nature of the world had not been recognized and the necessary language was nonexistent) and that the right categories to consider are actually the "derived" categories emerging at the moment? Would that be taking the nPOV too far? Or am I misinterpreting something again? Anyway that's just general philosophical talk where I try to get the big picture. So let me try to return to more concrete things. Mike uses the terminology "badly behaved", "not good" which I see a lot in this derived story. Instead of asking for a general definition of "badly behaved" maybe I should start by understanding some more examples first. In particular I would love to understand the example you mention of the quotient of a scheme by a group action. I agree that if the quotient does not exist than that's bad enough, but as you say there are cases when the quotient exists but "does not contain enough information". Can you tell me more precisely what is meant by that and how a "derived" quotient can fix these problems (or point me to a place where that is discussed)?

   Hi Urs, Zoran and Mike. Thanks for your answers!
   
   > Does that help?
   
   Quite a bit I should hope.
   
   So in particular from Mikes answer I get the feeling that "deriving" is not so much about seeing finer details, rather its about correcting defects the "traditional" categories had by replacing them with better behaved ones. (Let's for the sake of this discussion assume that all 1-categories are understood as $(\infty,1)$-categories in the natural way (i.e. lets say I've already been converted to the higher categorical point of view :-)). If from Zorans answer I may conclude that one cannot recover the traditional category from a derived version of it, would it then be correct (though somewhat radical) to say the following:
   the traditional categories people have been looking at (e.g. of schemes, of abelian sheaves etc) were wrong all along (mainly due to the fact that the higher categorical nature of the world had not been recognized and the necessary language was nonexistent) and that the right categories to consider are actually the "derived" categories emerging at the moment?
   Would that be taking the nPOV too far? Or am I misinterpreting something again?
   
   Anyway that's just general philosophical talk where I try to get the big picture.
   So let me try to return to more concrete things.
   
   Mike uses the terminology "badly behaved", "not good" which I see a lot in this derived story. Instead of asking for a general definition of "badly behaved" maybe I should start by understanding some more examples first. In particular I would love to understand the example you mention of the quotient of a scheme by a group action. I agree that if the quotient does not exist than that's bad enough, but as you say there are cases when the quotient exists but "does not contain enough information". Can you tell me more precisely what is meant by that and how a "derived" quotient can fix these problems (or point me to a place where that is discussed)?
6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 26th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexTypically in this business people realize that something is "bad" when it doesn't interact in the expected way with forming cohomology. For instance for some quotients of spaces by group actions, the resulting cohomology of the space becomes trivial, while really it should be the equivariant cohomology of the original space. This is recovered if instead of taking the naive quotient one takes the quotient as a stack, and then looks at the cohomology of that stack. But also notice that all things related to cohomology are really oo-categorical themselves, as described at [[cohomology]]. So what is really happening here is that given some 1-categorical objects, there is some _implicit_ way to think of them in an oo-categorical context to form their cohomology. Some other construction on that object, that might look like a natural thing to do, then actually turns out to be not compatible with that implicit identification. And then things look "bad". So as a rule of thumb: if you have a notion of cohomology on your objects that you care about, try to figure out in wich oo-category this cohomology is nothing but homotopy classes of morphisms. This then is a "good" oo-categorical context for your original objects.

   Typically in this business people realize that something is “bad” when it doesn’t interact in the expected way with forming cohomology.

   For instance for some quotients of spaces by group actions, the resulting cohomology of the space becomes trivial, while really it should be the equivariant cohomology of the original space. This is recovered if instead of taking the naive quotient one takes the quotient as a stack, and then looks at the cohomology of that stack.

   But also notice that all things related to cohomology are really oo-categorical themselves, as described at cohomology. So what is really happening here is that given some 1-categorical objects, there is some implicit way to think of them in an oo-categorical context to form their cohomology. Some other construction on that object, that might look like a natural thing to do, then actually turns out to be not compatible with that implicit identification. And then things look “bad”.

   So as a rule of thumb: if you have a notion of cohomology on your objects that you care about, try to figure out in wich oo-category this cohomology is nothing but homotopy classes of morphisms. This then is a “good” oo-categorical context for your original objects.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 26th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex> there are cases when the quotient exists but "does not contain enough information". Can you tell me more precisely what is meant by that ...? Well consider a group (Lie/topological/discrete/algebraic) acting on the point/terminal object. The quotient certainly exists, but looses all information about the group: all group actions on this object have the same quotient! However, we can consider the action groupoid, which in this case is an internal groupoid with 'one' object (i.e. the terminal object of the ambient category as the object of objects). This is none other than the delooping of the group, and this is the derived analogue of the classifying space construction in Top. It is better, because if you start with a finite group, you still get a finite groupoid, instead of an infinite dimensional space. The same with a fin. dim./compact Lie group: you get something very nice. But this isn't the end of the story: you now need to say what morphisms you have. In cases that I care about, this means using internal [[anafunctors]]: sort of like taking a resolution of the domain and then mapping that resolution to the codomain. An easy example of this is Cech cocycles, and they recover, up to equivalence, the same information contained in maps to classifying spaces. The big deal is now people are looking beyond internal groupoids to internal simplicial things, and so beyond categorical dimension 1, and so can capture more homotopy-like information in a sort of algebro-combinatorial way, especially for things that don't admit such descriptions, like algebraic geometry. (no singular simplices there!)

   there are cases when the quotient exists but “does not contain enough information”. Can you tell me more precisely what is meant by that …?

   Well consider a group (Lie/topological/discrete/algebraic) acting on the point/terminal object. The quotient certainly exists, but looses all information about the group: all group actions on this object have the same quotient! However, we can consider the action groupoid, which in this case is an internal groupoid with ’one’ object (i.e. the terminal object of the ambient category as the object of objects). This is none other than the delooping of the group, and this is the derived analogue of the classifying space construction in Top. It is better, because if you start with a finite group, you still get a finite groupoid, instead of an infinite dimensional space. The same with a fin. dim./compact Lie group: you get something very nice. But this isn’t the end of the story: you now need to say what morphisms you have. In cases that I care about, this means using internal anafunctors: sort of like taking a resolution of the domain and then mapping that resolution to the codomain. An easy example of this is Cech cocycles, and they recover, up to equivalence, the same information contained in maps to classifying spaces. The big deal is now people are looking beyond internal groupoids to internal simplicial things, and so beyond categorical dimension 1, and so can capture more homotopy-like information in a sort of algebro-combinatorial way, especially for things that don’t admit such descriptions, like algebraic geometry. (no singular simplices there!)

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeApr 26th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> the traditional categories people have been looking at (e.g. of schemes, of abelian sheaves etc) were wrong all along (mainly due to the fact that the higher categorical nature of the world had not been recognized and the necessary language was nonexistent) and that the right categories to consider are actually the "derived" categories emerging at the moment? I would disagree. A study of any object requires study of many associated objects. The associated objects should be constructed the most inherent way, and often this leads to infinity categories of various kinds. Categories of (quasi)coherent sheaves indeed have some information which their derived categories (of any kind: dg, A infinity, triangulated...) may not preserve: the cohomological information is preserved but some finite information is typically lost. The conditions of preservation sometimes have genuine geometrical nature. In noncommutative algebraic geometry situation is much worse, where the derived version is more or less the same kind as commutative, while at abelian level we are still knowing very little.

   the traditional categories people have been looking at (e.g. of schemes, of abelian sheaves etc) were wrong all along (mainly due to the fact that the higher categorical nature of the world had not been recognized and the necessary language was nonexistent) and that the right categories to consider are actually the “derived” categories emerging at the moment?

   I would disagree. A study of any object requires study of many associated objects. The associated objects should be constructed the most inherent way, and often this leads to infinity categories of various kinds. Categories of (quasi)coherent sheaves indeed have some information which their derived categories (of any kind: dg, A infinity, triangulated…) may not preserve: the cohomological information is preserved but some finite information is typically lost. The conditions of preservation sometimes have genuine geometrical nature. In noncommutative algebraic geometry situation is much worse, where the derived version is more or less the same kind as commutative, while at abelian level we are still knowing very little.

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeApr 26th 2010
   • (edited Apr 26th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItex> Typically in this business people realize that something is "bad" when it doesn't interact in the expected way with forming cohomology. More fundamentally: with forming certain limits or colimits. Both homology and cohomology are derived functors which hence measure (or to say differently: correct) obstructions to the left or right nonexactness.

   Typically in this business people realize that something is “bad” when it doesn’t interact in the expected way with forming cohomology.

   More fundamentally: with forming certain limits or colimits. Both homology and cohomology are derived functors which hence measure (or to say differently: correct) obstructions to the left or right nonexactness.

10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeApr 26th 2010
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    Author: zskoda
    Format: MarkdownItexAn interesting case are sheaves for Rosenberg's [[Q-topology]] where one considers families of cones or cocones which do not need to come from Grothendieck pre(co)topology. Such sheaves do not form a topos. I do not know what they do form (I mean it is not known if there is a weakening of Giraud's axioms which would abstractly characterize sheaves in Q-topology. It would be also interesting to see which kind of cohomology may be related to that generalization of sheaf theory (and also its higher categorical analogues).

    An interesting case are sheaves for Rosenberg’s Q-topology where one considers families of cones or cocones which do not need to come from Grothendieck pre(co)topology. Such sheaves do not form a topos. I do not know what they do form (I mean it is not known if there is a weakening of Giraud’s axioms which would abstractly characterize sheaves in Q-topology. It would be also interesting to see which kind of cohomology may be related to that generalization of sheaf theory (and also its higher categorical analogues).

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeApr 26th 2010
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    Author: Mike Shulman
    Format: MarkdownItexI agree with Zoran #8: it's not necessarily that the original categories were *wrong*, but that we also need to study other things. > Typically in this business people realize that something is "bad" when it doesn't interact in the expected way with forming cohomology. Yes, but here we can also interpret "cohomology" in the extreme nPOV viewpoint of "maps out of." In the examples we're talking about here, at least, we're forming "colimits" which are determined by maps out of them, i.e. by their "cohomology," so saying that their cohomology is wrong is saying the same thing as saying that they're the wrong colimit in the wrong category.

    I agree with Zoran #8: it’s not necessarily that the original categories were wrong, but that we also need to study other things.

    Typically in this business people realize that something is “bad” when it doesn’t interact in the expected way with forming cohomology.

    Yes, but here we can also interpret “cohomology” in the extreme nPOV viewpoint of “maps out of.” In the examples we’re talking about here, at least, we’re forming “colimits” which are determined by maps out of them, i.e. by their “cohomology,” so saying that their cohomology is wrong is saying the same thing as saying that they’re the wrong colimit in the wrong category.

12. • CommentRowNumber12.
    • CommentAuthorMichael_Bachtold
    • CommentTimeApr 27th 2010
    • (edited Apr 27th 2010)
    • PermaLink
    Author: Michael_Bachtold
    Format: MarkdownItexThanks again for all your answers. It will take me some time to digest whats been said, specially to read and understand the nLab page on cohomology. But for the moment I want to ask something more basic: is it actually safe to consider the theory of 1-categories as a special case of the theory of $(\infty,1)$-categories? More precisely: the nerve construction allows one to view a 1-category as an $(\infty,1)$-category and is a fully faithful functor. That seems like a good start. Now as I understood, every construction from classical categories has an analogous construction in higher categories. Do these constructions commute with the nerve? E.g. limits: if I have a diagram in a classical category, take its limit and then view the result as something higher categorical, do I get the same answer as when I first view things higher categorically and then take the limit in the $(\infty,1)$ sense? And about topoi: if I view a classical topos as an $(\infty,1)$-category, does it satisfy the axioms of a higher topos? I.e. are classical topoi instances of higher topoi?

    Thanks again for all your answers. It will take me some time to digest whats been said, specially to read and understand the nLab page on cohomology. But for the moment I want to ask something more basic: is it actually safe to consider the theory of 1-categories as a special case of the theory of $(\infty,1)$-categories? More precisely: the nerve construction allows one to view a 1-category as an $(\infty,1)$-category and is a fully faithful functor. That seems like a good start. Now as I understood, every construction from classical categories has an analogous construction in higher categories.

    Do these constructions commute with the nerve?

    E.g. limits: if I have a diagram in a classical category, take its limit and then view the result as something higher categorical, do I get the same answer as when I first view things higher categorically and then take the limit in the $(\infty,1)$ sense?

    And about topoi: if I view a classical topos as an $(\infty,1)$-category, does it satisfy the axioms of a higher topos? I.e. are classical topoi instances of higher topoi?

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2010
    • (edited Apr 27th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItexVery good questions. I put answers to these now in the corresponding nLab-entries, to the extent that they are not there yet. But just quickly: > if I have a diagram in a classical category, take its limit and then view the result as something higher categorical, do I get the same answer as when I first view things higher categorically and then take the limit in the $(\infty,1)$-sense? Taking a limit _in_ a category yields the same result as first regarding the category as a special (oo,1)-category and then taking the oo-limit _in_ that oo-category. > And about topoi: if I view a classical topos as an $(\infty,1)$-category, does it satisfy the axioms of a higher topos? I.e. are classical topoi instances of higher topoi? This requires some care: a 1-category of sheaves, regarded as an (oo,1)-category, is never an (oo,1)-topos.But still the following is true: the (oo,1)-category of "1-localic (oo,1)-toposes" (those that are oo-categories of oo-sheaves on ordinary 1-categories) and [[geometric morphism]]s between them is equivalent to the 1-category of 1-toposes and geometric morphisms between them. So the thing is that while sheaves on a site and oo-stacks on the same site are very different, under geometric morphisms both will just remember the ordinary topos on the site. In particular, notice that every ordinary topos on a site is the full sub-(oo,1)-category of the oo-topos of oo-stacks on that site on 0-[[truncated]] objects: an oo-stack that happens to take values in oo-groupoids that are just sets is precisely the same thing as just an ordinary sheaf. I'll put more details on this into [[(n-1)-topos]] in a moment (not much there right now except a complaint by Mike on a previous rushed version of a stub of this entry :-)

    Very good questions. I put answers to these now in the corresponding nLab-entries, to the extent that they are not there yet. But just quickly:

    if I have a diagram in a classical category, take its limit and then view the result as something higher categorical, do I get the same answer as when I first view things higher categorically and then take the limit in the $(\infty,1)$-sense?

    Taking a limit in a category yields the same result as first regarding the category as a special (oo,1)-category and then taking the oo-limit in that oo-category.

    And about topoi: if I view a classical topos as an $(\infty,1)$-category, does it satisfy the axioms of a higher topos? I.e. are classical topoi instances of higher topoi?

    This requires some care: a 1-category of sheaves, regarded as an (oo,1)-category, is never an (oo,1)-topos.But still the following is true: the (oo,1)-category of “1-localic (oo,1)-toposes” (those that are oo-categories of oo-sheaves on ordinary 1-categories) and geometric morphisms between them is equivalent to the 1-category of 1-toposes and geometric morphisms between them.

    So the thing is that while sheaves on a site and oo-stacks on the same site are very different, under geometric morphisms both will just remember the ordinary topos on the site. In particular, notice that every ordinary topos on a site is the full sub-(oo,1)-category of the oo-topos of oo-stacks on that site on 0-truncated objects: an oo-stack that happens to take values in oo-groupoids that are just sets is precisely the same thing as just an ordinary sheaf.

    I’ll put more details on this into (n-1)-topos in a moment (not much there right now except a complaint by Mike on a previous rushed version of a stub of this entry :-)

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexokay, I expanded [[(n,1)-topos]] a little, stating the crucial property about how they relate for different $n$. Not enough yet, but need to interrupt to get so,me dinner... :-)

    okay, I expanded (n,1)-topos a little, stating the crucial property about how they relate for different $n$. Not enough yet, but need to interrupt to get so,me dinner… :-)

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeApr 27th 2010
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex> the nerve construction allows one to view a 1-category as an (∞,1)-category and is a fully faithful functor It's true that 1-categories can be viewed as $(\infty,1)$-categories, but I think it's misleading to call this the "nerve." If we choose quasicategories as our model for $(\infty,1)$-categories, then this construction is modeled by the nerve, but in other models it's something completely different. > the (oo,1)-category of "1-localic (oo,1)-toposes" (those that are oo-categories of oo-sheaves on ordinary 1-categories) and geometric morphisms between them is equivalent to the 1-category of 1-toposes and geometric morphisms between them. I think you probably mean the (2,1)-category of 1-toposes and geometric morphisms.

    the nerve construction allows one to view a 1-category as an (∞,1)-category and is a fully faithful functor

    It’s true that 1-categories can be viewed as $(\infty,1)$-categories, but I think it’s misleading to call this the “nerve.” If we choose quasicategories as our model for $(\infty,1)$-categories, then this construction is modeled by the nerve, but in other models it’s something completely different.

    the (oo,1)-category of “1-localic (oo,1)-toposes” (those that are oo-categories of oo-sheaves on ordinary 1-categories) and geometric morphisms between them is equivalent to the 1-category of 1-toposes and geometric morphisms between them.

    I think you probably mean the (2,1)-category of 1-toposes and geometric morphisms.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItex> I think you probably mean the (2,1)-category of 1-toposes and geometric morphisms. Yes, right.

    I think you probably mean the (2,1)-category of 1-toposes and geometric morphisms.

    Yes, right.

17. • CommentRowNumber17.
    • CommentAuthorHarry Gindi
    • CommentTimeMay 1st 2010
    • (edited May 1st 2010)
    • PermaLink
    Author: Harry Gindi
    Format: Markdown@Zoran, by the way, Reid Barton showed on MO that there is no nontrivial abelian category that is also a topos (this follows from the axioms pretty quickly).

    @Zoran, by the way, Reid Barton showed on MO that there is no nontrivial abelian category that is also a topos (this follows from the axioms pretty quickly).

18. • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 1st 2010
    • (edited May 1st 2010)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexWell, naturally. In a topos, $X \times 0 \cong 0$ and of course $X \times 1 \cong X$. So $0 \cong 1$ would imply $0 \cong X$ for every $X$. On the other hand $0 \cong 1$ in an abelian category. In some sense this is the heart of the difference. Define the theory AT as the Horn theory whose axioms are universal Horn sentences satisfied by all toposes and by all abelian categories. The point is that there are plenty of such sentences: the sentence that all epis are regular, all monos are regular, that the pullback of an epi is an epi, the pushout of a mono is a mono, and many others. Then (hope I get this right), the universal Horn sentences satisfied by all abelian categories are exactly those provable in the Horn theory AT + (axiom that $0 \cong 1$), and the universal Horn sentences satisfied by all toposes are exactly those provable in the Horn theory AT + (axiom that $X \times 0 \cong 0$). Quite a remarkable result! See <a href="http://cs.nyu.edu/pipermail/fom/1998-January/001029.html">this discussion</a> by Vaughan Pratt, which harks back to a discussion between Freyd and Pratt <a href="http://www.mta.ca/~cat-dist/catlist/1999/atcat">here</a>.

    Well, naturally. In a topos, $X \times 0 \cong 0$ and of course $X \times 1 \cong X$. So $0 \cong 1$ would imply $0 \cong X$ for every $X$. On the other hand $0 \cong 1$ in an abelian category.

    In some sense this is the heart of the difference. Define the theory AT as the Horn theory whose axioms are universal Horn sentences satisfied by all toposes and by all abelian categories. The point is that there are plenty of such sentences: the sentence that all epis are regular, all monos are regular, that the pullback of an epi is an epi, the pushout of a mono is a mono, and many others. Then (hope I get this right), the universal Horn sentences satisfied by all abelian categories are exactly those provable in the Horn theory AT + (axiom that $0 \cong 1$), and the universal Horn sentences satisfied by all toposes are exactly those provable in the Horn theory AT + (axiom that $X \times 0 \cong 0$). Quite a remarkable result! See this discussion by Vaughan Pratt, which harks back to a discussion between Freyd and Pratt here.

19. • CommentRowNumber19.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2010
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexThere is a certain amount of sloppiness in the description of AT in # 18 (since one has to carefully specify signatures to even make sense of it), which I will try to atone for by writing up some Lab articles on things like Horn clauses. Also I need to spend some time understanding Freyd's result properly, but this would also make a nice addition to the Lab.

    There is a certain amount of sloppiness in the description of AT in # 18 (since one has to carefully specify signatures to even make sense of it), which I will try to atone for by writing up some Lab articles on things like Horn clauses. Also I need to spend some time understanding Freyd’s result properly, but this would also make a nice addition to the Lab.

20. • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2010
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexTodd, that is indeed a very interesting result, which I didn't know about! It should definitely be labbified. Although at least glancing at Freyd's first email, I feel that talking about toposes is misleading -- the result seems to really be about pretoposes, which makes it less surprising to me, since pretoposes and abelian categories are both essentially defined by exactness properties. Both have the same unary exactness -- they are (Barr-)[[exact categories]] -- so they mainly differ in their treatment of coproducts: pretoposes are [[extensive category|extensive]] while abelian categories are [[additive category|additive]]. (In the pretopos case this is the definition -- a pretopos is an exact extensive category -- but probably it is not quite enough in the abelian case? Is any exact additive category abelian?) And apparently also any category satisfying AT must be a product of an abelian category and a pretopos!

    Todd, that is indeed a very interesting result, which I didn’t know about! It should definitely be labbified. Although at least glancing at Freyd’s first email, I feel that talking about toposes is misleading – the result seems to really be about pretoposes, which makes it less surprising to me, since pretoposes and abelian categories are both essentially defined by exactness properties. Both have the same unary exactness – they are (Barr-)exact categories – so they mainly differ in their treatment of coproducts: pretoposes are extensive while abelian categories are additive. (In the pretopos case this is the definition – a pretopos is an exact extensive category – but probably it is not quite enough in the abelian case? Is any exact additive category abelian?) And apparently also any category satisfying AT must be a product of an abelian category and a pretopos!

21. • CommentRowNumber21.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2010
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItex> the result seems to really be about pretoposes I'm a little bit murky on this point myself. I do think of this result as being mainly about finite exactness properties. > Is any exact additive category abelian? Isn't it? > And apparently also any category satisfying AT must be a product of an abelian category and a pretopos! Yes, that is really incredible! At least on first impression.

    the result seems to really be about pretoposes

    I’m a little bit murky on this point myself. I do think of this result as being mainly about finite exactness properties.

    Is any exact additive category abelian?

    Isn’t it?

    And apparently also any category satisfying AT must be a product of an abelian category and a pretopos!

    Yes, that is really incredible! At least on first impression.

22. • CommentRowNumber22.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2010
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexThere is a sense in which abelian categories are almost extensive, and again it hinges on the difference of behavior of 0. For in an abelian category, coproducts are disjoint, and while coproducts are not stable under pullback, pushouts are. So the only thing missing is that the empty coproduct is not stable under pullback.

    There is a sense in which abelian categories are almost extensive, and again it hinges on the difference of behavior of 0. For in an abelian category, coproducts are disjoint, and while coproducts are not stable under pullback, pushouts are. So the only thing missing is that the empty coproduct is not stable under pullback.

23. • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexMight it be that one useful way to think about the fact that toposes and abelian categories are closely related while still crucially different is in terms of [[Goodwillie calculus]]?

    Might it be that one useful way to think about the fact that toposes and abelian categories are closely related while still crucially different is in terms of Goodwillie calculus?

24. • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2010
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexIs every epic (resp. monic) in an exact additive category necessarily a cokernel (resp. a kernel)? The remark about the behavior of 0 reminds me a bit of Daniel Schaeppi's recent [proof](http://mathoverflow.net/questions/14402/semiadditivity-and-dualizability-of-2) that if a symmetric monoidal category has a zero object and $I+I$ is dualizable, then the category is semiadditive. Somehow additivity seems to happen very quickly once you have a few "additive-like" properties.

    Is every epic (resp. monic) in an exact additive category necessarily a cokernel (resp. a kernel)?

    The remark about the behavior of 0 reminds me a bit of Daniel Schaeppi’s recent proof that if a symmetric monoidal category has a zero object and $I+I$ is dualizable, then the category is semiadditive. Somehow additivity seems to happen very quickly once you have a few “additive-like” properties.

25. • CommentRowNumber25.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2010
    • (edited May 2nd 2010)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexOkay, I guess I'll quote from Cats and Alligators, 1.594 (p. 90): > $\mathbf{A}$ is abelian iff it is an effective regular additive category. Although it may not be universally true that Freyd-Scedrov use terms in their standard sense, I think it's true here. I would transcribe the argument here, except that I find this book a pain-in-the-neck to read on occasion. :-)

    Okay, I guess I’ll quote from Cats and Alligators, 1.594 (p. 90):

    $\mathbf{A}$ is abelian iff it is an effective regular additive category.

    Although it may not be universally true that Freyd-Scedrov use terms in their standard sense, I think it’s true here. I would transcribe the argument here, except that I find this book a pain-in-the-neck to read on occasion. :-)

26. • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2010
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexOkay, well that seems like it's probably true then. I think it's easy to see that it's true for monics: in an additive category, any monic $i\colon A\hookrightarrow B$ gives rise to an equivalence relation on $B$ via $A\oplus B \rightrightarrows B$ which are the second projection and the sum of the second projection with $i$, and if this equivalence relation is effective, then $A$ is the kernel (in the additive sense) of its quotient. I can imagine that there is some cleverer argument for the other case.

    Okay, well that seems like it’s probably true then. I think it’s easy to see that it’s true for monics: in an additive category, any monic $i\colon A\hookrightarrow B$ gives rise to an equivalence relation on $B$ via $A\oplus B \rightrightarrows B$ which are the second projection and the sum of the second projection with $i$, and if this equivalence relation is effective, then $A$ is the kernel (in the additive sense) of its quotient. I can imagine that there is some cleverer argument for the other case.

27. • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeJun 25th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexIn this 4-year-old discussion we all gave a lot of explanation of why one might want to pass from 1-categories to (∞,1)-categories. But we never actually answered the original question, which I will paraphrase as > Does the (∞,1)-functor which embeds an abelian 1-category into its derived (∞,1)-category satisfy a universal property? That is, we gave a lot of reasons why one may want to embed a 1-category into *some* (∞,1)-category, which might have been the question that Michael was actually "really" wondering about. But we didn't say much about how one might characterize that *particular* embedding. Michael didn't originally specify exactly which "derived category" he meant, but to my mind the most natural one to ask the question about is the unbounded one. Does it have a universal property? It feels as though it ought to, since it can be constructed by first looking at simplicial objects (which are equivalent to nonnegatively-graded chain complexes by Dold-Kan) and then stabilizing. But what exactly is the universal property? In particular, I am wondering to what extent it depends on having enough projectives or injectives.

    In this 4-year-old discussion we all gave a lot of explanation of why one might want to pass from 1-categories to (∞,1)-categories. But we never actually answered the original question, which I will paraphrase as

    Does the (∞,1)-functor which embeds an abelian 1-category into its derived (∞,1)-category satisfy a universal property?

    That is, we gave a lot of reasons why one may want to embed a 1-category into some (∞,1)-category, which might have been the question that Michael was actually “really” wondering about. But we didn’t say much about how one might characterize that particular embedding.

    Michael didn’t originally specify exactly which “derived category” he meant, but to my mind the most natural one to ask the question about is the unbounded one. Does it have a universal property? It feels as though it ought to, since it can be constructed by first looking at simplicial objects (which are equivalent to nonnegatively-graded chain complexes by Dold-Kan) and then stabilizing. But what exactly is the universal property?

    In particular, I am wondering to what extent it depends on having enough projectives or injectives.

28. • CommentRowNumber28.
    • CommentAuthorZhen Lin
    • CommentTimeJun 25th 2014
    • PermaLink
    Author: Zhen Lin
    Format: MarkdownItexIs there a way of at least functorially recovering the image of the embedding? I think "homology concentrated in degree 0" is not a property preserved by stable functors.

    Is there a way of at least functorially recovering the image of the embedding? I think “homology concentrated in degree 0” is not a property preserved by stable functors.

29. • CommentRowNumber29.
    • CommentAuthorFosco
    • CommentTimeJun 26th 2014
    • (edited Jun 26th 2014)
    • PermaLink
    Author: Fosco
    Format: MarkdownItex> Michael didn't originally specify exactly which "derived category" he meant, but to my mind the most natural one to ask the question about is the unbounded one. Does it have a universal property? HA.1.3.3 ("The Universal property of $D^-(\mathcal{A})$") tells us that we must prefer the (right bounded) one; see in particular Thm 1.3.3.8! ![enter image description here][1] [1]: http://i.stack.imgur.com/X64UE.png

    Michael didn’t originally specify exactly which “derived category” he meant, but to my mind the most natural one to ask the question about is the unbounded one. Does it have a universal property?

    HA.1.3.3 (“The Universal property of $D^-(\mathcal{A})$”) tells us that we must prefer the (right bounded) one; see in particular Thm 1.3.3.8!

    

30. • CommentRowNumber30.
    • CommentAuthorZhen Lin
    • CommentTimeJun 26th 2014
    • PermaLink
    Author: Zhen Lin
    Format: MarkdownItexSo, in words, the full subcategory of the derived category consisting of those chain complexes whose homology is concentrated in non-negative degrees has the universal property of the $(\infty, 1)$-category obtained freely adding homotopy colimits for simplicial objects that are degreewise projective. That sounds reasonable (even unsurprising); unfortunately, $\mathcal{D}^{-}_{\ge 0} (\mathcal{A})$ is not stable, and $\mathcal{A}_{proj}$ is not equivalent to $\mathcal{A}$!

    So, in words, the full subcategory of the derived category consisting of those chain complexes whose homology is concentrated in non-negative degrees has the universal property of the $(\infty, 1)$-category obtained freely adding homotopy colimits for simplicial objects that are degreewise projective. That sounds reasonable (even unsurprising); unfortunately, $\mathcal{D}^{-}_{\ge 0} (\mathcal{A})$ is not stable, and $\mathcal{A}_{proj}$ is not equivalent to $\mathcal{A}$!

31. • CommentRowNumber31.
    • CommentAuthorMike Shulman
    • CommentTimeJun 26th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexYeah, Zhen about sums it up. I find that universal property very unsatisfying.

    Yeah, Zhen about sums it up. I find that universal property very unsatisfying.