User eric a. bunch - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:15:36Z http://mathoverflow.net/feeds/user/3664 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92633/significance-of-the-vanishing-of-k-1a Significance of the vanishing of $K_{-1}(A)$ Eric A. Bunch 2012-03-30T00:40:32Z 2012-03-30T22:31:22Z <p>In M. Schlichting's <a href="http://homepages.warwick.ac.uk/~masiap/research/frob5.pdf" rel="nofollow">paper</a>, he defines the negative $K$-theory for derived categories. In this he states that for $\mathcal{A}$ an idempotent complete (see below) triangulated category, $K_{-1}(\mathcal{A}) = 0$ is equivalent to the fact that for any full triangle embedding $\mathcal{A} \hookrightarrow \mathcal{B}$ of $\mathcal{A}$ into an idempotent complete triangulated category $\mathcal{B}$, we have that $\mathcal{B}/\mathcal{A}$ is also idempotent complete.</p> <p>An additive category is said to be idempotent complete (sometimes called Karoubian) if for every idempotent $e: A \rightarrow A, \hspace{2pt} e^2 = e$ defines a splitting $A = Im(e)\oplus Ker(e)$. </p> <p>My question is, what is the importance of this quotient category $\mathcal{B}/\mathcal{A}$ being idempotent complete?</p> <p>I would even be happy if the following, more specific question were answered. For a ring R, Bass defined the negative $K$-groups to be the cokernel of the map </p> <p>$K_{-(n-1)}(R[t]) \oplus K_{-(n-1)}(R[t^{-1}]) \rightarrow K_{-(n-1)}(R[t, t^{-1}])$, for $n \geq 0$. </p> <p>If $R$ is regular, by the homotopy invariance property of the $K$-groups, $K_{-1}(R) = 0$. It is proven in Schlichting's paper that the negative $K$-groups for a ring defined this way coincide with the negative $K$-groups of the category $Ch^b(Proj(R))$ of bounded chain complexes of the exact category of projective $R$-modules. The more specific question is what is the significance of the above quotient condition where $\mathcal{A}$ is the category $Ch^b(Proj(R))$?</p> http://mathoverflow.net/questions/83363/is-there-a-nice-application-of-category-theory-to-functional-complex-harmonic-ana/83490#83490 Answer by Eric A. Bunch for Is there a nice application of category theory to functional/complex/harmonic analysis? Eric A. Bunch 2011-12-15T02:08:29Z 2011-12-15T02:08:29Z <p>At the suggestion of Yemon, I have moved my comment to an answer. The <a href="http://en.wikipedia.org/wiki/Gelfand_representation" rel="nofollow">Gelfand representation</a> gives an equivalence between the category of commutative, unital $C^*$-algebras and the opposite category of compact Hausdorff spaces. </p> <p>Breifly, let $A$ be a $C^*$-algebra, and let $\Sigma$ be the collection of nonzero homomorphisms $A \rightarrow \mathbb{C}$. Then $\Sigma$ sits inside $A^*$, the dual of $A$. Thus we can endow it with the weak-$*$ topology. With this topology, if we consider $C(\Sigma)$, the algebra of continuous functions $\Sigma \rightarrow \mathbb{C}$, it turns out that we obtain a canonical isometric $*$-isomorphism $A \rightarrow C(\Sigma)$. </p> <p>The functor $CommC^*Alg \rightarrow CptHdTop^{op}$ given by $A \mapsto \Sigma$ defined above is an equivalence of categories. A great reference for the details is Conway's book on functional analysis (but he doesn't mention categories or functors). </p> http://mathoverflow.net/questions/24773/why-do-categorical-foundationalists-want-to-escape-set-theory Why do categorical foundationalists want to escape set theory? Eric A. Bunch 2010-05-15T16:20:01Z 2011-06-02T15:08:36Z <p>This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.</p> <p>I know that it's possible to formulate category theory within set theory while still being albe to construct the useful things one would want from category theory. So as far as I understand, all normal mathematics that involves category theory can be done as long as a little caution is taken. </p> <p>I also know that some people (categorical foundationalsists) would still like to formulate category theory without use of or reference to set theory. While I admit that I am curious about this for curiosity's sake, I'm not sure if there are any practical motivations for doing this. The only reason for wanting to separate category theory from set theory that I have read about is for the sake of autonomy of category theory'.</p> <p>So my question is twofold: What other reasons might categorical foundationalists have for separating category theory from set theory, and what practical purposes might it serve to do this?</p> http://mathoverflow.net/questions/21785/topos-with-alternate-subobject-lattice Topos' with alternate subobject lattice? Eric A. Bunch 2010-04-18T23:11:39Z 2011-03-02T00:13:19Z <p>We know that for any topos E, and for any object A in E, the subobjects of A, Sub(A), form a Heyting lattice.</p> <p>Does anybody know of any sort of modification of the definition of a topos that makes Sub(A) a different type of lattice? Could we get an incomplete lattice, or maybe a quantum lattice?</p> <p>I'm curious because I know a lot(all?) of logical systems can be realized as a lattice, and I think this may be an interesting way to look at some alternative logics.</p> http://mathoverflow.net/questions/29552/intuition-for-the-satellite-of-a-functor Intuition for the satellite of a functor Eric A. Bunch 2010-06-25T22:44:56Z 2010-12-09T11:01:33Z <p>Occasionally in math I come across constructions or tools that are a bit convoluted. I can look at these constructions and see that they indeed perform the task they were made to do, but sometimes I can't see why they $\textit{should}$ perform this task; why is that the logical thing to choose for the job? Right now, I'm having this issue with the satellite of a functor.</p> <p>Just to recall, given an additive functor between two abelian categories $F:\mathcal{C} \rightarrow \mathcal{D}$, the satellite is another functor $S_-^1(F): \mathcal{C} \rightarrow \mathcal{D}$ defined by </p> <p>$S_-^1(F)(M) = lim(ker(F(M) \rightarrow F(P)))$</p> <p>where $0 \rightarrow M \rightarrow P \rightarrow N \rightarrow 0$ is an exact sequence with $P$ a projective object. Then a derived functor is formed by taking iterations of the satellite: $S_-^n(F) = S_-(S_-^{n-1}(F))$. More information can be found on <a href="http://ncatlab.org/nlab/show/satellite" rel="nofollow">nlab</a>.</p> <p>I am learning about derived functors in a slightly different setting, namely with nonadditive categories where there are not necessarily enough projectives in the category $\mathcal{C}$, and so the definition is modified slightly; perhaps the definition is more transparent in the standard setting. </p> <p>So my question is</p> <blockquote> <p>Is there any intuition for why the satellite is the correct tool to use for obtaining derived functors? If I needed to create a derived functor out of a given functor, is there a logical progression that would lead me to define the satellite?</p> </blockquote> http://mathoverflow.net/questions/33427/roadmap-to-a-proof-of-the-atiyah-singer-index-theorem-which-uses-k-theory Roadmap to a proof of the Atiyah-Singer Index Theorem which uses K-Theory Eric A. Bunch 2010-07-26T18:16:09Z 2010-07-27T09:57:44Z <p>Lately my studies have been focusing on learning the machinery of K-Theory, and I thought that learning the Atiyah-Singer Index Theorem would be a good way to see K-Theory in action a bit and to learn a deep result on the way. From what I have read, there are a few methods of proof of Atiyah-Singer, one of which uses K-Theory. Also from what I have read, it seems that I have most of the background knowledge to approach the proof of Atiyah-Singer. </p> <p>However, it doesn't seem that there is a standard reference or sequence of references to go to in order to learn the proof of this theorem. In particular, I am not sure which book(s) would be best to look at to see a proof of Atiyah-Singer which utilizes K-Theory. I have found a few that seem to take the K-Theory approach to the theorem, but I have no way of telling how good or useful they are.</p> <p>So what I am asking for is a roadmap or a reference to a proof of the Atiyah-Singer Index Theorem that uses K-Theory, and of course other advice concerning learning this theorem is welcome as well.</p> http://mathoverflow.net/questions/22065/algebraic-properties-of-the-algebra-of-continuous-functions-on-a-manifold Algebraic properties of the algebra of continuous functions on a manifold. Eric A. Bunch 2010-04-21T15:18:42Z 2010-04-25T01:02:16Z <blockquote> <p>Does the algebra of continuous functions from a compact manifold to $\mathbb{C}$ satisfy any specific algebraic property?</p> </blockquote> <p>I'm not sure what kind of algebraic property I expect, but I feel that because of the Gel'fand transform, it may not be unreasonable to expect something. We can drop the compactness condition if we switch to continuous functions to $\mathbb{C}$ that vanish at infinity.</p> <p>I'm really hoping for some necessary and sufficient condition, but if anybody knows of any sort of condition, that would be appreciated.</p> http://mathoverflow.net/questions/83363/is-there-a-nice-application-of-category-theory-to-functional-complex-harmonic-ana/83490#83490 Comment by Eric A. Bunch Eric A. Bunch 2011-12-16T00:16:53Z 2011-12-16T00:16:53Z @Yemon: Definitely, reference to the morphisms is necessary to fully realize the categorical viewpoint. My category theorist friend would be horrified to see me make no mention of what the functor does on morphisms! I hope he can forgive me if he ever reads this. http://mathoverflow.net/questions/83363/is-there-a-nice-application-of-category-theory-to-functional-complex-harmonic-ana Comment by Eric A. Bunch Eric A. Bunch 2011-12-15T02:19:00Z 2011-12-15T02:19:00Z @Yemon: My comment has been transferred to an answer. Questions like this concerning category theory are always tricky to ask as well as answer. I think the power of category theory comes from its usefulness as an organizational tool, and that it gives an easier, more structured way to look at the big picture. I feel like the best way to see the usefulness of category theory in these terms is by just seeing a lot of examples. http://mathoverflow.net/questions/83363/is-there-a-nice-application-of-category-theory-to-functional-complex-harmonic-ana/83490#83490 Comment by Eric A. Bunch Eric A. Bunch 2011-12-15T02:13:27Z 2011-12-15T02:13:27Z As far as I understand, this is one of the jumping off points in certain non-commutative geometry/pseudo-geometry programs. When studying non-commutative C*-algebras, the thought is that they should correspond to some non-commutative space analogous to the commutative case. http://mathoverflow.net/questions/83363/is-there-a-nice-application-of-category-theory-to-functional-complex-harmonic-ana Comment by Eric A. Bunch Eric A. Bunch 2011-12-15T00:38:39Z 2011-12-15T00:38:39Z This is a pretty well-known example, but I like too much to go without mentioning it. The Gelfand representation gives an equivalence between the category of commutative, unital C*-algebras and the opposite category of compact Hausdorff spaces. http://mathoverflow.net/questions/24773/why-do-categorical-foundationalists-want-to-escape-set-theory/66692#66692 Comment by Eric A. Bunch Eric A. Bunch 2011-06-02T18:53:58Z 2011-06-02T18:53:58Z I was not worrying about the foundations of category theory for no purpose; at a certain point working with categories, it was necessary for me to fix a universe and work with that. I had never had to do this before, and it led me to look a bit more into foundations of category theory and mathematics founded only on category theory. I'm perfectly happy to stop with fixing a universe, but if there were additional advantages to be had by being more careful with category theory, then I didn't want to miss out. http://mathoverflow.net/questions/60322/references-on-lie-groups-and-dynamical-systems Comment by Eric A. Bunch Eric A. Bunch 2011-04-01T23:48:44Z 2011-04-01T23:48:44Z Also, this question should probably be made community wiki. http://mathoverflow.net/questions/60322/references-on-lie-groups-and-dynamical-systems Comment by Eric A. Bunch Eric A. Bunch 2011-04-01T23:47:54Z 2011-04-01T23:47:54Z I am also interested in this connection. Searching the internet and literature hasn't been too forthcoming yet, but I have only just begun. It seems to me that a likely connection would be through representations of amenable groups. Jaoby, have you found anything useful in this direction on your own? http://mathoverflow.net/questions/33427/roadmap-to-a-proof-of-the-atiyah-singer-index-theorem-which-uses-k-theory/33428#33428 Comment by Eric A. Bunch Eric A. Bunch 2010-07-26T22:01:57Z 2010-07-26T22:01:57Z Thanks! It seems I was confused and/or misinformed. I did not know their paper used K-Theory. http://mathoverflow.net/questions/29552/intuition-for-the-satellite-of-a-functor/29830#29830 Comment by Eric A. Bunch Eric A. Bunch 2010-07-05T05:30:53Z 2010-07-05T05:30:53Z I agree with you that your answer answers the question of `why derived functors' pretty well. Thanks again for your answer, although I was hoping for something specific about the satellite. http://mathoverflow.net/questions/29552/intuition-for-the-satellite-of-a-functor/29830#29830 Comment by Eric A. Bunch Eric A. Bunch 2010-06-29T15:21:37Z 2010-06-29T15:21:37Z Thanks for the answer. I'll have to think about this a bit more; I've never worked with Kan extensions before. http://mathoverflow.net/questions/29552/intuition-for-the-satellite-of-a-functor Comment by Eric A. Bunch Eric A. Bunch 2010-06-26T03:36:26Z 2010-06-26T03:36:26Z Yes, thanks. The different setting in which I'm learning about derived functors also does not require additivity of the functor or abelian categories. It has been changed. http://mathoverflow.net/questions/24773/why-do-categorical-foundationalists-want-to-escape-set-theory/24783#24783 Comment by Eric A. Bunch Eric A. Bunch 2010-05-16T16:02:28Z 2010-05-16T16:02:28Z Thanks again! Very helpful. I had requested the article through interlibrary loan, but it takes a bit to process. http://mathoverflow.net/questions/24773/why-do-categorical-foundationalists-want-to-escape-set-theory/24783#24783 Comment by Eric A. Bunch Eric A. Bunch 2010-05-16T00:16:39Z 2010-05-16T00:16:39Z Todd, I know about Lawvere's article because I've seen it referenced almost everywhere, but I can't seem to get a hold of it: I feel out of the loop. Is there a particular place you know of that I could find it? Also, I second unknown's request about the article by Kreisel :) http://mathoverflow.net/questions/24773/why-do-categorical-foundationalists-want-to-escape-set-theory/24783#24783 Comment by Eric A. Bunch Eric A. Bunch 2010-05-15T19:24:35Z 2010-05-15T19:24:35Z @Todd: Thanks for the answer! Do you know anything about foundations using the category of categories? I've read that this is another alternative, but that it is problematic. However, I've had the toughest time finding anything that actually explains in detail using the category of categories as a foundation. Does your answer still hold true for this approach to foundations? http://mathoverflow.net/questions/22065/algebraic-properties-of-the-algebra-of-continuous-functions-on-a-manifold/22151#22151 Comment by Eric A. Bunch Eric A. Bunch 2010-04-28T14:41:47Z 2010-04-28T14:41:47Z @Yemon: I'm afraid I don't know anything about spectral triples(other than what wikipedia tells me), but thanks for the suggestion nevertheless. I'm sorry my question was not very precise and therefore frustrating. I guess some properties don't translate well. Thanks again to everybody for all the suggestions!