most recent 30 from http://mathoverflow.net 2013-05-23T01:04:03Z http://mathoverflow.net/feeds/user/22002 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120410/why-are-subextensions-of-galois-extensions-also-galois Mozibur Ullah 2013-01-31T13:52:20Z 2013-02-01T13:43:30Z

<p>Generally a Galois extension is defined to be an algebraic extension that is also normal &amp; separable. It is then shown that in the sequence of field extensions $L|M|K$ if $L|K$ is Galois then $L|M$ is. This follows since the same property is valid for separable &amp; normal extensions individually. It also follows that $L|K$ is a Galois extension iff the set of elements of $L$ invariant under the action of $Aut_K L$ is $K$</p> <p>In Robalo Delgados thesis on <a href="https://dspace.ist.utl.pt/bitstream/2295/575330/1/dissertacao.pdf" rel="nofollow">Galois Categories</a> referenced in <a href="http://ncatlab.org/nlab/show/Grothendieck%27s+Galois+theory" rel="nofollow">nLab-Grothendiecks Galois Theory</a> he takes the opposite tack, and in <em>definition 3.2.1.1</em> defines an algebraic extension of fields $L|K$ to be a Galois extension iff the set of elements of $L$ invariant under the action of $Aut_K L$ is $K$.</p> <p>It is then shown that in the sequence of algebraic field extensions $L|M|K$ if $L|K$ is Galois then $L|M$ is. This is asserted to be an obvious deduction (and so has no details), I don't see the obviousness...can someone clarify.</p> <p>In <em>proposition 3.2.1.3</em> he shows that Galois extension is normal and separable.</p> <p>All this appears to be in the opposite order of the standard treatments. One reason I'm interested in his formulation, if it is correct, is that one side of the Galois correspondence follows easily from this.</p> <p><em>disclaimer</em>: I've already asked this question on <a href="http://math.stackexchange.com/questions/290882/why-are-subextensions-of-galois-extensions-also-galois" rel="nofollow">math.stackexchange</a> but the answers there revolved around characterising Galois extensions as being normal &amp; separable, and then showing this property follows.</p>

http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation/120026#120026 Mozibur Ullah 2013-01-27T13:56:44Z 2013-01-27T15:32:59Z

<p>I think economics is far more closely connected with the body politic than it with mathematics. Also applying mathematics to economics is also a political act and a signification. (Mathematics has an association with permanance which can be used symbolically to shore up a certain contingent political/economic order).</p> <p>Physics examines the world by supposing the physical world follows a rational order, and that by dint of effort this order is discoverable. I can't see how this applies to the social order of societie(s); how does one measure wealth, imagination, violence, ethics, power, desire, criminality? </p> <p>Whereas mathematics applied to physics captures something of its fundamental relationships, it appears to me that a mathematical model of a social order can only captures superficial and contingent things. </p>

http://mathoverflow.net/questions/119732/what-is-the-characteristic-property-of-surjective-submersions Mozibur Ullah 2013-01-24T06:39:53Z 2013-01-24T23:19:12Z

<p>In Lee's 'Introduction to smooth manifolds' he states that given smooth manifolds X,Y and a surjective submersion f:X→Y, then f is a smoothly final map, that is for any further smooth manifold Z, and any map g:Y→Z, we have g smooth iff g∘f is smooth.</p> <p>He then says that problem 4.7 shows why this property is 'characteristic'. I can't see why the reverse implication should hold.</p> <p>Unfortunately, google-books doesn't show that page, nor do I have access to a mathematical library, can some-one enlighten me as to what he means?</p> <p>One of the answers to this <a href="http://mathoverflow.net/questions/115145/are-immersions-submersions-captured-in-generalised-smooth-spaces-by-some-universa" rel="nofollow">question</a> states a characteristic property, but it doesn't appear on the face of it what Lee has in mind.</p>

http://mathoverflow.net/questions/94354/essential-reads-in-the-philosophy-of-mathematics-and-set-theory/116297#116297 Mozibur Ullah 2012-12-13T17:06:50Z 2012-12-13T17:06:50Z

<p>Borges 'The Library of Babel' is a beautiful meditation on all sorts of philosophical positions around the 'idea' of infinity, epistemology, the sociology of science, set theory paradoxes. Its literature &amp; not philosophy though, or is it the other way around...?</p>

http://mathoverflow.net/questions/115540/what-are-the-current-possibilities-for-infinite-dimensional-manifolds Mozibur Ullah 2012-12-05T20:13:28Z 2012-12-05T20:13:28Z

<p>According to wikipedia, by a theorem of Henderson '69, infinite-dimensional <a href="http://en.wikipedia.org/wiki/Fr%C3%A9chet_manifold" rel="nofollow">Frechet Manifolds</a> embed as open subspaces of Hilbert Space. They need to be seperable &amp; metric. They are generalisations of <a href="http://en.wikipedia.org/wiki/Banach_manifold" rel="nofollow">Banach Manifolds</a>, so they too have the same property.</p> <p>Michor &amp; Krigel, say 'this does not make them [Banach Manifolds] interesting [enough]' in <a href="http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf" rel="nofollow">The convenient setting of Global Analysis</a>, </p> <p>What are the other directions to take, when looking for interesting infinite-dimensional manifolds, (besides the one Michor/Kriegel outline in their book). And has a canonical choice begun to establish itself yet?</p>

http://mathoverflow.net/questions/69251/is-mac-lane-still-the-best-place-to-learn-category-theory/115284#115284 Mozibur Ullah 2012-12-03T12:39:47Z 2012-12-03T12:39:47Z

<p>I found the Catsters on youtube divinely useful.</p> <p>John Baez, in his not so weekly blog, inspiring</p> <p>The n-category cafe, to keep you going</p> <p>Eugenia Chengs notes on category theory was tremendously useful</p> <p>Eventually, Maclane began to make sense, as did Borceaux; but oh, ever so slowly</p> <p>Sets for mathematicians is pretty</p> <p>And the n-lab is a great resource, but mostly dazzles my eyes...</p> <p>And yes, 1-category theory is definitely best to start with, and be familiar with; but keep an eye on the higher grounds too.</p>

http://mathoverflow.net/questions/115209/are-verma-modules-universally-characterised Mozibur Ullah 2012-12-02T21:35:04Z 2012-12-03T08:18:31Z

<p>I'm having trouble understanding the definition of Verma Module in wikipedia. It later goes to show that it satisfies what appears to be a universal property (which I'm also having trouble understanding - the page is a notational mess). Surely this can be taken as the definition? And is there a clear way of expressing it?</p>

http://mathoverflow.net/questions/115199/are-exotic-spheres-still-exotic-in-generalised-smooth-spaces Mozibur Ullah 2012-12-02T21:08:40Z 2012-12-03T07:09:24Z

<p>This is really more of a philosophical question, and the title is somewhat rhetorical:</p> <p>Exotic spheres are a feature of smooth manifold theory, where certain spheres can have more than one differentiable structure. In a generalised smooth space, say for definiteness a diffeological space, its trivial to construct different diffelogies on the space.</p> <p>Are then exotic spheres really just an artifact of using the wrong smooth technology?</p> <p>I guess what I'm getting is that the surprise was that there were manifolds with more than one smooth structure, one could argue that this was a first pointer that we should work in a category where the manifolds could be given different smooth structures, since the natural &amp; usual one couldn't maintain uniqueness of smooth structure for certain manifolds.</p>

http://mathoverflow.net/questions/115146/where-should-one-go-to-learn-about-triangulated-categories Mozibur Ullah 2012-12-02T04:13:32Z 2012-12-02T22:03:32Z

<p>Lurie's book, higher topos theory describes a new notion of a triangulated category, which is apparently much more natural than the usual definition. Obviously by now a great deal of work has been done on triangulated categories, has anyone translated any of this work into this new language? </p> <p>That is should one go straight to his book, or become reasonably acquainted with triangulated categories first, in the hope that by then people will have become more familiar with that language by then?</p>

http://mathoverflow.net/questions/115145/are-immersions-submersions-captured-in-generalised-smooth-spaces-by-some-universa Mozibur Ullah 2012-12-02T04:08:22Z 2012-12-02T21:48:46Z

<p>Immersions &amp; sumersions are important in differential manifolds. They rely on their definition of the construction of the tangent bundle.</p> <p>I realise that generalised smooth spaces do not have a canonical tangent bundle. </p> <p>But they have better categorical properties, but nastier objects. Is it possible to define what a submerssion/immersion is here by a universal property?</p>

http://mathoverflow.net/questions/115201/can-a-classifying-space-be-characterised-universally Mozibur Ullah 2012-12-02T21:14:41Z 2012-12-02T21:34:45Z

<p>I'm having trouble understanding what classifying spaces are in general.</p> <p>It seems to me, that they are terminal in a category of bundles whose morphisms are pullback squares, is this correct?</p>

http://mathoverflow.net/questions/115187/is-the-stochastic-integral-useful-outside-of-financial-mathematics Mozibur Ullah 2012-12-02T17:02:46Z 2012-12-02T17:05:22Z

<p>The lesbegue integral and its generalisations are useful in many places in mathematics other than its place of origin, e.g., representation theory. Is the same true for the stochastic integral? From my meager understanding, it seems to rely on a similar theoretical background, but stochastically interpreted. Or is it too new an innovation to have gained ground elsewhere?</p>

http://mathoverflow.net/questions/115099/are-grothendieck-universes-enough-for-the-foundations-of-category-theory Mozibur Ullah 2012-12-01T19:23:44Z 2012-12-01T22:53:46Z

<p>Grothendieck universes are equivalent to ZFC+a strongly inaccessible cardinal. This is low on the large cardinal axiom list. Is it enough to place category theory on a firm foundational basis, and how about higher category theory, does it remain enough?</p>

http://mathoverflow.net/questions/115091/what-notion-captures-the-class-of-all-classes Mozibur Ullah 2012-12-01T17:45:09Z 2012-12-01T22:28:37Z

<p>In ZFC there is no set that is the set of all sets, for this we introduce the notion of class. But then what is the 'class' of all classes, is it actually a class? Do we apply the same idea again? But then at what stage do we stop? Does this show that classes are not the right notion to go beyond sets, but more of an ad-hoc solution?</p> <p>Further, within foundational category theory, we have the notion of grothendieck universes, if i recall rightly, this is equivalent to introducing an axiom that an inaccessible cardinal exists. Does this subsume, or is equivalent to the notion of classes?</p> <p>Finally, what is the formalism that uses classes to extend ZFC, is the NBG?</p>

http://mathoverflow.net/questions/115097/when-is-a-sum-of-idempotents-idempotent-in-commutative-ring-theory Mozibur Ullah 2012-12-01T18:56:23Z 2012-12-01T18:56:23Z

<p>As this <a href="http://mathoverflow.net/questions/115067/necessary-and-sufficient-conditions-for-a-sum-of-idempotents-to-be-idempotent" rel="nofollow">question</a> demonstrates that the sum of idempotents is idempotent iff every pairwise product is zero, for finite matrices with complex entries.</p> <p>What additional restrictions do we need to put in for this to remain true in commutative ring theory? Of course the same trivial direction remains trivial, it just that the other direction , I would hazard restricts what kinds of rings we can use. </p>

http://mathoverflow.net/questions/108128/how-do-we-avoid-circularity-when-we-build-a-structure-for-zfc Mozibur Ullah 2012-09-26T02:35:34Z 2012-09-26T08:27:27Z

<p>when investigating ZFC as a formal language a structure is a set, are we not engaging in circular logic here? Or is 'set' thought of in a more primitive sense?</p>

http://mathoverflow.net/questions/107857/is-there-a-universal-property-that-characterises-the-join-of-two-categories Mozibur Ullah 2012-09-22T18:52:13Z 2012-09-22T19:32:43Z

<p>Let A &amp; B be two categories, the join A*B is created by stipulating its class of object is the disjoint union of the objects of A &amp; B, the morphisms remain the 'same', but we throw in an extra morphism for every object a in A, and b in B.</p> <p>that is:</p> <p>A*B[a,a']=A[a,a'] if a,a' are in A</p> <p>A*B[b,b']=B[b,b'] if b,b' are in B</p> <p>A*B[a,b]=1 if a is in A, and b in B</p> <p>A*B[b,a]=0 if b is in B, and a in A</p> <p>it seems like a pretty ad-hoc construction (its obviously based on a construction coming from algebraic topology), is there a more categorical way of defining this?</p>

http://mathoverflow.net/questions/107846/is-there-any-connection-between-the-consistent-histories-interpretation-of-quantu Mozibur Ullah 2012-09-22T14:35:37Z 2012-09-22T14:35:37Z

<p>Kripke semantics interpret intuitionistic logic by a partially ordered set of worlds/situations. Consistent histories interpretation of QM elaborates the copenhagen interpretation where a consistent set of histories can be assigned probabilities. </p> <p>I don't really understand either of these formalisms, but it does appear on the face of it that the formalisms should have something in common, do they?</p>

http://mathoverflow.net/questions/107818/what-categories-correspond-to-the-typed-lambda-calculus-with-parametric-types Mozibur Ullah 2012-09-22T05:31:54Z 2012-09-22T13:23:47Z

<p>the unadorned typed lambda calculus correspond to the closed cartesian categories, but if we add in dependent or parametric types how are they then characterised?</p>

http://mathoverflow.net/questions/107230/is-there-a-category-of-topological-like-spaces-that-forms-a-topos Mozibur Ullah 2012-09-15T03:26:40Z 2012-09-15T12:08:46Z

<p>The category of <a href="http://ncatlab.org/nlab/show/convergence+space" rel="nofollow">convergence spaces</a> generalise topological spaces and form a quasi-topos, as topoi are allegedly nicer is there a nicer kind of topological-like space, the category of which forms a topos?</p>

http://mathoverflow.net/questions/103141/what-is-the-functor-tensor-product Mozibur Ullah 2012-07-26T01:16:32Z 2012-07-27T17:56:37Z

<p>I'm familiar with the tensor product of modules, but I've also come across functor tensor product (in emily riehls paper on homotopy limits), what are they, and how are they (if they are) related to traditional tensor products? (Emily shows that they can be defined as a particular coend, but that doesn't really provide any intuition for me). </p>

http://mathoverflow.net/questions/102005/what-is-a-de-vries-algebra Mozibur Ullah 2012-07-11T23:57:53Z 2012-07-12T10:01:01Z

<p>I've come across a set of <a href="http://people.maths.ox.ac.uk/hap/GuramBezSlides.pdf" rel="nofollow">slides</a> by Guram Bezhanishvili where he claims the category of compact hausdorff spaces is related by duality to de Vries algebras. What are they?</p>

http://mathoverflow.net/questions/101719/why-is-the-tangent-groupoid-useful Mozibur Ullah 2012-07-08T23:51:16Z 2012-07-09T04:30:58Z

<p>Let M be a smooth manifold. The classical construction is the tangent bundle TM. What does the tangent groupoid of M give me that this construction doesn't, and what is the best way of describing it? (Also, what is the standard notation for it, or do they just overload TM?)</p>

http://mathoverflow.net/questions/92974/what-is-the-dunford-integral-and-why-is-it-useful Mozibur Ullah 2012-04-03T08:06:37Z 2012-04-03T13:38:20Z

<p>Wikipedia <a href="http://en.wikipedia.org/wiki/Pettis_integral" rel="nofollow">http://en.wikipedia.org/wiki/Pettis_integral</a> defines the Pettis Integral for Banach space valued functions wrt to some measure space by duality. </p> <p>It calls the Pettis &amp; Bochner integral the weak &amp; strong integral respectively, which implies some kind of relationship; also, apparently there is a Dunford integral which specialises to the Pettis integral. </p> <p>My question is: Why are these weak integrals useful, what is the definition of the Dunford Integral and how does it specialise to the Pettis one, and what is the relationship between the Pettis Integral &amp; the Bochner Integral?.</p> <p>I've just noted: <a href="http://mathoverflow.net/questions/47721/weak-and-strong-integration-of-vector-valued-functions" rel="nofollow">http://mathoverflow.net/questions/47721/weak-and-strong-integration-of-vector-valued-functions</a>, which answers the part of my question about the Pettis Integral.</p>

http://mathoverflow.net/questions/90778/gluing-bundles-as-a-2-colimit Mozibur Ullah 2012-03-10T00:57:09Z 2012-03-14T12:31:43Z

<p>Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?</p>

http://mathoverflow.net/questions/90778/gluing-bundles-as-a-2-colimit/91169#91169 Mozibur Ullah 2012-03-14T12:31:43Z 2012-03-14T12:31:43Z

<p>not my answer, but David Carchedi's answer in a comment:</p> <p>'What you might be thinking is, the category of principal bundles over a fixed base is a 2-colimit over all covers of the base (or some cofinal subset) of the categories of principal bundles over that fixed base which trivialize over the given cover'</p>

http://mathoverflow.net/questions/90839/grothendieck-on-topological-vector-spaces/90870#90870 Mozibur Ullah 2012-03-11T02:34:55Z 2012-03-11T02:34:55Z

<p>These kind of statements are made from time, not just within subfields of mathematics, but also within the larger world. From painting is dead (I'm not sure who said this) &amp; history is dead (Fukuyama). </p>

http://mathoverflow.net/questions/90656/are-banach-manifolds-intrinsically-interesting Mozibur Ullah 2012-03-09T03:57:02Z 2012-03-09T14:33:59Z

<p>In the introduction to 'A convenient setting for Global Analysis', Michor &amp; Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open subsets of the modeling space for many modeling spaces."</p> <p>But finite-dimensional manifolds are found to be interesting even though they can be embedded in some euclidean space (of larger dimension). (Actually this seems to me, to make the above claim intuitively plausible, so that claim should be no more than we should expect).</p> <p>But they do go on to say that "Banach manifolds are not suitable for many questions of Global Analysis, as...a Banach Lie group acting effectively on a finite dimensional smooth manifold it must be finite dimensional itself.", which does seem a rather strong limitation. </p>

http://mathoverflow.net/questions/120410/why-are-subextensions-of-galois-extensions-also-galois/120439#120439 Mozibur Ullah 2013-02-01T13:41:08Z 2013-02-01T13:41:08Z

@Mueller: I'm beginning to suspect that Delgado is wrong in his claim, particularly the 'immediacy' of the deduction...

http://mathoverflow.net/questions/120410/why-are-subextensions-of-galois-extensions-also-galois Mozibur Ullah 2013-01-31T18:52:11Z 2013-01-31T18:52:11Z

@Brandenburg: ditto.

http://mathoverflow.net/questions/120410/why-are-subextensions-of-galois-extensions-also-galois Mozibur Ullah 2013-01-31T18:51:59Z 2013-01-31T18:51:59Z

@Quid: Yes, this is what I was getting at.

http://mathoverflow.net/questions/120410/why-are-subextensions-of-galois-extensions-also-galois Mozibur Ullah 2013-01-31T17:23:31Z 2013-01-31T17:23:31Z

@Tveiten: I have, and I still don't think its answered my question. They go via the route of characterising Galois extensions first as algebraic, separable &amp; normal extensions - and then show the property that Delgado uses to characterise a Galois extension follows. Whereas Delgado starts of with this and deduces the characterisation.

http://mathoverflow.net/questions/119980/concise-model-of-modern-fiat-money-and-its-non-conservation/120026#120026 Mozibur Ullah 2013-01-27T15:28:25Z 2013-01-27T15:28:25Z

By denying that it's answerable in its own terms? When Newton wrote about his law of gravitation, he made certain he could model both time and space mathematically, the <i>background</i> to his physics. I'm denying that this background is available in economics. Theories don't exist in a vacuum , they exist in a larger theoretical space.

http://mathoverflow.net/questions/85096/arithmetic-derivative Mozibur Ullah 2013-01-25T10:37:05Z 2013-01-25T10:37:05Z

I don't think Vitale deserved the brushoff that the moderators have given him. I certainly think the arithmatic derivative is an intriguing idea, and Vitale should have been encouraged to phrase his question in a more sensible fashion: such as has there been any interesting results shown/proved/reformulated.

http://mathoverflow.net/questions/119732/what-is-the-characteristic-property-of-surjective-submersions/119791#119791 Mozibur Ullah 2013-01-25T01:58:34Z 2013-01-25T01:58:34Z

and your additional comment is useful too.

http://mathoverflow.net/questions/119732/what-is-the-characteristic-property-of-surjective-submersions/119791#119791 Mozibur Ullah 2013-01-25T01:54:13Z 2013-01-25T01:54:13Z

great, going by Lees answer I see that my question wasn't quite right. But I am interested in how I phrased it. Do you think it can actually hold locally? I've accepted Lees answer as it only seems fair since I picked up the question from his book. But your answer is equally worthwhile. It doesn't seem quite correct that one should choose.

http://mathoverflow.net/questions/119732/what-is-the-characteristic-property-of-surjective-submersions/119793#119793 Mozibur Ullah 2013-01-25T01:49:29Z 2013-01-25T01:49:29Z

I wasn't expecting the author of the text to turn up! Thanks, what you had in mind wasn't what I was expecting, but I see now I should have done, its exactly analogous to final maps in Top.

http://mathoverflow.net/questions/115099/are-grothendieck-universes-enough-for-the-foundations-of-category-theory Mozibur Ullah 2013-01-22T04:22:35Z 2013-01-22T04:22:35Z

@Shulman: I wouldn't expect a different foundation to play exactly the same role as the original, but one would expect there to be a great deal of commonality. Does this mean that ETCS is entirely standalone? So although its inspired by category theory that scaffolding can be taken away. I find this point a little confusing: why do this? Isn't there an 'elementary theory of categories' that doesn't rely on ZFC (I thought Maclane tries to do this in his book) then why not keep to the natural order of 'inspiration'? Unless of course there is no such elementary theory as it runs into difficulties

http://mathoverflow.net/questions/107230/is-there-a-category-of-topological-like-spaces-that-forms-a-topos/107249#107249 Mozibur Ullah 2013-01-22T04:10:47Z 2013-01-22T04:10:47Z

I just reread this, and realised this is the kind of answer that I was looking for.

http://mathoverflow.net/questions/51136/what-are-galois-categories-used-for Mozibur Ullah 2012-12-25T19:08:35Z 2012-12-25T19:08:35Z

Galois categories inspired the Tannakian categories formalism that reconstructs an affine group scheme from its finite-dimensional representations.

http://mathoverflow.net/questions/90839/grothendieck-on-topological-vector-spaces/90870#90870 Mozibur Ullah 2012-12-21T02:11:49Z 2012-12-21T02:11:49Z

@Joel:Thanks for clarifying Fukuyamas title. I thought Fukuyama was also stating if not explicitly, then implicitly that liberal democracies were the endpoint of the evolution of political forms of a state? Quite, except when the &quot;I&quot; is an influential person in the field, remarks such as these are more influential and drive people away.

http://mathoverflow.net/questions/115209/are-verma-modules-universally-characterised Mozibur Ullah 2012-12-09T17:28:36Z 2012-12-09T17:28:36Z

@Humphreys: Thanks for the online reference. I do realise that a universal construction is only for characterisation, and automatically proves uniqueness, so long as existence is shown. I have online access, but not a useful library access, unfortunately.

http://mathoverflow.net/questions/115209/are-verma-modules-universally-characterised Mozibur Ullah 2012-12-09T17:23:24Z 2012-12-09T17:23:24Z

@Shanmukha_Srinivasan: My silence has more to with getting a job than anything else:). It would probably have been better to have waited until my circumstances were a bit more stable. The question popped into my head sometime after I first learnt about Lie Groups/Algebras, and I ran across Verma Module on Wikipedia, and that in itself is a while ago. (I used Dragon Milicic online notes which I found very useful). I agree with Tom Leinsters comments too.