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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeAug 28th 2014
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   Author: Fosco
   Format: MarkdownItexLet me elaborate a bit on the crude question: it is said in the page [[abelian category]] that the presence of a (epi,mono) factorization in a preabelian $\mathcal{C}$ should entail that it is indeed abelian. No counterexample has been found for this conjecture, as far as I proceeded to read the discussion here on the nforum. Given my (our) recent <a href="http://nforum.mathforge.org/discussion/6157/tstructures-and-factorization-systems/">divertissements</a> with factorization systems in relation with t-structures, and given the tight connection between these and abelian categories, I would like to understand better what's going on here. The heart of a t-structure, as defined in <a href="http://www1.mat.uniroma1.it/people/fiorenza/tstruct.pdf">our</a> 4.1 is an $\infty$-category whose heart is abelian. Given that stable $\infty$-categories should be "the things in higher world such that their homotopy category is triangulated", what should a good definition of "the thing whose homotopy category is abelian" be? To my eye it seems really unlikely that all "respectable" abelian categories can be characterized as hearts of t-structures in stable $\infty$-categories: is it true, is it false?

   Let me elaborate a bit on the crude question: it is said in the page abelian category that the presence of a (epi,mono) factorization in a preabelian $\mathcal{C}$ should entail that it is indeed abelian. No counterexample has been found for this conjecture, as far as I proceeded to read the discussion here on the nforum. Given my (our) recent divertissements with factorization systems in relation with t-structures, and given the tight connection between these and abelian categories, I would like to understand better what’s going on here.

   The heart of a t-structure, as defined in our 4.1 is an $\infty$-category whose heart is abelian. Given that stable $\infty$-categories should be “the things in higher world such that their homotopy category is triangulated”, what should a good definition of “the thing whose homotopy category is abelian” be? To my eye it seems really unlikely that all “respectable” abelian categories can be characterized as hearts of t-structures in stable $\infty$-categories: is it true, is it false?

2. • CommentRowNumber2.
   • CommentAuthordomenico_fiorenza
   • CommentTimeAug 29th 2014
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   Author: domenico_fiorenza
   Format: MarkdownItexAn $\infty$-category $\mathbf{C}$ such that i) $\mathbf{C}(X,Y)$ is a homotopy discrete infinity loop space for any $X, Y$, i.e., there exists an infinite sequence of (well behaved) topological spaces $Z_0, Z_1,Z_2$, with $Z_0\cong \mathbf{C}(X,Y)$ and together with homotopy equivalences $Z_i\cong \Omega Z_{i+1}$ for any $i\geq 0$, such that $\pi_n Z_0=0$ for any $n\geq 1$. This is the homotopy theory version of $\mathbf{C}(X,Y)$ being an abelian group. For instance, if the abelian group is $\mathbb{Z}$, then the corresponding homotopy discrete infinity loop space is the Eilenberg-MacLane spectrum $\mathbb{Z},K(\mathbb{Z},1), K(\mathbb{Z},2),\dots$; ii) $\mathbf{C}$ has a 0 object, (homotopy) kernels, cokernels and biproducts iii) the natural morphism from the coimage to the image is an equivalence The homotopy category of such a $\mathbf{C}$ is an abelian category (note that $\mathbf{C}(X,Y)$ being homotopically discrete is necessary in order that kernels and cokernels in $\mathbf{C}$ do induce kernels and cokernels in $h\mathbf{C}$). Concerning the last question, the answer is "it is (almost) true", see Damien Calaque's answer <a href="http://mathoverflow.net/a/105825/8320">here</a>

   An $\infty$-category $\mathbf{C}$ such that

   i) $\mathbf{C}(X,Y)$ is a homotopy discrete infinity loop space for any $X, Y$, i.e., there exists an infinite sequence of (well behaved) topological spaces $Z_0, Z_1,Z_2$, with $Z_0\cong \mathbf{C}(X,Y)$ and together with homotopy equivalences $Z_i\cong \Omega Z_{i+1}$ for any $i\geq 0$, such that $\pi_n Z_0=0$ for any $n\geq 1$. This is the homotopy theory version of $\mathbf{C}(X,Y)$ being an abelian group. For instance, if the abelian group is $\mathbb{Z}$, then the corresponding homotopy discrete infinity loop space is the Eilenberg-MacLane spectrum $\mathbb{Z},K(\mathbb{Z},1), K(\mathbb{Z},2),\dots$;

   ii) $\mathbf{C}$ has a 0 object, (homotopy) kernels, cokernels and biproducts

   iii) the natural morphism from the coimage to the image is an equivalence

   The homotopy category of such a $\mathbf{C}$ is an abelian category (note that $\mathbf{C}(X,Y)$ being homotopically discrete is necessary in order that kernels and cokernels in $\mathbf{C}$ do induce kernels and cokernels in $h\mathbf{C}$).

   Concerning the last question, the answer is “it is (almost) true”, see Damien Calaque’s answer here

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeAug 29th 2014
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   Author: Mike Shulman
   Format: MarkdownItexDomenico, doesn't your (i) actually imply that C is equivalent to hC? It seems to me that this is not the right question: stability is already the infinity-analogue of abelianness for 1-categories.

   Domenico, doesn’t your (i) actually imply that C is equivalent to hC?

   It seems to me that this is not the right question: stability is already the infinity-analogue of abelianness for 1-categories.

4. • CommentRowNumber4.
   • CommentAuthordomenico_fiorenza
   • CommentTimeAug 29th 2014
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   Author: domenico_fiorenza
   Format: MarkdownItexHi Mike, yes, (i) is the statement that $\mathbf{C}(X,Y)\to \pi_0\mathbf{C}(X,Y)$ is an equivalence, together with the fact that $\pi_0\mathbf{C}(X,Y)$ is an abelian group. Concerning stability vs. abeliannness, I completely agree: stability is the right infinity-analogue, no doubt. Yet, a stable $\infty$-category fails to satisfy (iii) above, and so its homotopy category is not abelian, but only triangulated in general. So if one wants that $\mathbf{C}$ is such that $h\mathbf{C}$ is abelian one has to ask something different from stability, which is (or at least should be (i-iii) above). One may wonder whether examples of this naturally exist, apart the tautological "an $\infty$-category which is equivalent to an abelian category. A nice example is the heart of a t-structure on a $\infty$-stable category. In Higher Algebra this result is only stated, and only for the homotopy category $h\mathbb{C}$, addressing the reader to the classical Beilinson-Bernstein-Deligne Asterisque for the proof. But it is actually not hard to work out a direct proof within the context of t-structures on $\infty$-stable categories.

   Hi Mike,

   yes, (i) is the statement that $\mathbf{C}(X,Y)\to \pi_0\mathbf{C}(X,Y)$ is an equivalence, together with the fact that $\pi_0\mathbf{C}(X,Y)$ is an abelian group. Concerning stability vs. abeliannness, I completely agree: stability is the right infinity-analogue, no doubt. Yet, a stable $\infty$-category fails to satisfy (iii) above, and so its homotopy category is not abelian, but only triangulated in general. So if one wants that $\mathbf{C}$ is such that $h\mathbf{C}$ is abelian one has to ask something different from stability, which is (or at least should be (i-iii) above). One may wonder whether examples of this naturally exist, apart the tautological “an $\infty$-category which is equivalent to an abelian category. A nice example is the heart of a t-structure on a $\infty$-stable category. In Higher Algebra this result is only stated, and only for the homotopy category $h\mathbb{C}$, addressing the reader to the classical Beilinson-Bernstein-Deligne Asterisque for the proof. But it is actually not hard to work out a direct proof within the context of t-structures on $\infty$-stable categories.

5. • CommentRowNumber5.
   • CommentAuthorFosco
   • CommentTimeAug 29th 2014
   • (edited Aug 29th 2014)
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   Author: Fosco
   Format: MarkdownItex> Stability is already the infinity-analogue of abelianness for 1-categories. If you want I can put it in another form: which features have in common the hearts of t-structures in stable $\infty$-categories? I'm aware of what you say (thank you for having pointed out this!), and yet it seems quite unsatisfactory not to make a distinction. I would like to understand where is my mistake, if any. Abelian categories "should" arise as hearts of stable $\infty$-categories; these hearts are, in principle, a $\infty$-categorical object *whose homotopy category* is abelian; when you kill higher informations in a stable $\mathcal{C}$ you obtain a triangulated category and from a $t$-structure on it you obtain its heart. I can accept that maybe abelian and triangulated categories *both* stem from stable categories; but then again, only *some* stable $\infty$-categories will have an abelian homotopy category. > One may wonder whether examples of this naturally exist, apart the tautological This is precisely what I was about to write :) I saw that we were commenting at the same time.

   Stability is already the infinity-analogue of abelianness for 1-categories.

   If you want I can put it in another form: which features have in common the hearts of t-structures in stable $\infty$-categories? I’m aware of what you say (thank you for having pointed out this!), and yet it seems quite unsatisfactory not to make a distinction. I would like to understand where is my mistake, if any.

   Abelian categories “should” arise as hearts of stable $\infty$-categories; these hearts are, in principle, a $\infty$-categorical object whose homotopy category is abelian; when you kill higher informations in a stable $\mathcal{C}$ you obtain a triangulated category and from a $t$-structure on it you obtain its heart. I can accept that maybe abelian and triangulated categories both stem from stable categories; but then again, only some stable $\infty$-categories will have an abelian homotopy category.

   One may wonder whether examples of this naturally exist, apart the tautological

   This is precisely what I was about to write :) I saw that we were commenting at the same time.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 30th 2014
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   Author: Mike Shulman
   Format: MarkdownItexWhat is the point of distinguishing between an abelian 1-category and an $(\infty,1)$-category that is equivalent to it?

   What is the point of distinguishing between an abelian 1-category and an $(\infty,1)$-category that is equivalent to it?

7. • CommentRowNumber7.
   • CommentAuthordomenico_fiorenza
   • CommentTimeAug 30th 2014
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   Author: domenico_fiorenza
   Format: MarkdownItexIndeed the notion of ablian $\infty$-category as defined above is not very interesting per se. Rather such an object can naturally arise as a full subcategory of a stable $\infty$-category, and that's the whole point. In other words, if $\mathbf{C}$ is the heart of a t-structure in a stable $\infty$-category $\mathbf{D}$, then $\mathbf{C}$ is equivalent to $h\mathbf{C}$ and $h\mathbf{C}$ is an abelian category. A way of saying these two things at the same time is to say that $\mathbf{C}$ satisfies (i-iii).

   Indeed the notion of ablian $\infty$-category as defined above is not very interesting per se. Rather such an object can naturally arise as a full subcategory of a stable $\infty$-category, and that’s the whole point. In other words, if $\mathbf{C}$ is the heart of a t-structure in a stable $\infty$-category $\mathbf{D}$, then $\mathbf{C}$ is equivalent to $h\mathbf{C}$ and $h\mathbf{C}$ is an abelian category. A way of saying these two things at the same time is to say that $\mathbf{C}$ satisfies (i-iii).

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeAug 31st 2014
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   Author: Mike Shulman
   Format: MarkdownItexIn other words, the answer to Fosco's question "which features have in common the hearts of t-structures in stable ∞-categories?" is just "they are abelian (1-)categories".

   In other words, the answer to Fosco’s question “which features have in common the hearts of t-structures in stable ∞-categories?” is just “they are abelian (1-)categories”.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeAug 31st 2014
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   Author: Mike Shulman
   Format: MarkdownItexBy the way, it seems to me that your (i-iii) are a little odd, and possibly not quite right. The [[biproducts]] in (ii) already ensure that $C$ is enriched over abelian monoids, but you haven't said anything to specify that this agrees with the abelian group structure on homsets coming from (i). It would automatically if the latter were actually an $Ab$-enrichment, i.e. the composition is bilinear, but you haven't specified that either.

   By the way, it seems to me that your (i-iii) are a little odd, and possibly not quite right. The biproducts in (ii) already ensure that $C$ is enriched over abelian monoids, but you haven’t said anything to specify that this agrees with the abelian group structure on homsets coming from (i). It would automatically if the latter were actually an $Ab$-enrichment, i.e. the composition is bilinear, but you haven’t specified that either.

10. • CommentRowNumber10.
    • CommentAuthordomenico_fiorenza
    • CommentTimeAug 31st 2014
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    Author: domenico_fiorenza
    Format: MarkdownItexRight, I've been too naive in (i-iii). The idea there was to write out what are the features of the heart of a t-structure in a stable $\infty$-category that then imply that its image in the homotopy category is an 1-abelian category. So, yes, they are no way optimal or complete and need to be refined in order to be an actual definition. However, how you remark, there is no point in giving such an abstract definition: in the end one is just speaking of $\infty$-categories which are equivalent to a abelian (1-)categories.

    Right, I’ve been too naive in (i-iii). The idea there was to write out what are the features of the heart of a t-structure in a stable $\infty$-category that then imply that its image in the homotopy category is an 1-abelian category. So, yes, they are no way optimal or complete and need to be refined in order to be an actual definition. However, how you remark, there is no point in giving such an abstract definition: in the end one is just speaking of $\infty$-categories which are equivalent to a abelian (1-)categories.