User aleph0 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:09:44Z http://mathoverflow.net/feeds/user/1681 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51777/is-an-a-infinity-thing-the-same-the-same-as-strict-thing-viewed-through-a-homotpy Is an A-infinity thing the same the same as strict thing viewed through a homotpy equivalence? aleph0 2011-01-11T17:27:51Z 2011-01-19T06:41:36Z <p>If I have a topological monoid ($X,\mu$), a space $Y$ and a homotopy equivalence $f,g$ between them, then $Y$ has the strucutre of an $A_\infty$ space defined 'pointwise' by $$y_1 * y_2 := g \left(\mu(f(y_1),f(y_2))\right)$$ Also, I have heard someone say the reverse as "Every $A_\infty$-space is (weakly?) homotopy equivalent to a topological monoid."</p> <p>My question is this: In how far is this the general case? Can I define/think of $A_\infty$ structures as strict structures as viewed through the distorting glasses of a homotopy equivalence? When is an $A_\infty$ structure of this type - i.e. is there always an equivalent strict version? (I guess no in general. Can you tell me more?) </p> <p>FYI: I started out with the question: Is the based loop space of a space $X$ always on the other side of a homotopy equivalence of a (strict) topological group?</p> <p>thanks, a.</p> http://mathoverflow.net/questions/22295/1-vs-category-weakly-enriched-over-spaces (∞,1) vs Category weakly enriched over spaces aleph0 2010-04-23T03:40:15Z 2010-05-23T12:02:52Z <p>What is the difference between:</p> <p>($\infty,1$) categories - in which have for two objects you have an ($\infty,0$) category of morphisms (i.e. a space of morphisms) </p> <p>and</p> <p>categories weakly enriched over spaces - by that I mean categories such that hom(x,y) is always a space and composition is defined only up to (coherent) homotopy</p> <p>?</p> http://mathoverflow.net/questions/430/homological-algebra-for-commutative-monoids/8637#8637 Answer by aleph0 for Homological Algebra for Commutative Monoids? aleph0 2009-12-12T01:38:51Z 2009-12-12T01:38:51Z <p>I used to think about this problem in relation to a chain theory for <strong>bordism</strong> ( as mentioned by Josh Shadlen above).<BR> The problems you have with monoids is first and foremost that the category is not balanced. That means that you can have an epimorphism that is also a monomorphism but NOT an isomorphism. eg. the inclusion of N --> Z. <BR> Subsequently most constructions that you would like to make - notably short exact sequences and the snake lemma - fail at some level.<BR> I made a few notes on this as part of my investigation into bordism theory / homework assignment <a href="http://www.sirarnold.net/maths/homework_bordism_and_homological_algebra_for_monoids.pdf" rel="nofollow"> here</a>. <BR> In this case, we have complexes of free abelian monoids whose homology takes it's values in abelian groups and yet, the long exact sequence does not come from a short exact sequence of monoid complexes.<BR> <BR> My references include:<BR><BR> [Bau89] Friedrich W. Bauer. Generalised homology theories and chain complexes. Annali di Mathematica pura ed applicata, CLV:143–191, 1989.<BR><BR> [Bau95] Friedrich W. Bauer. Bordism theories and chain complexes. Journal of Pura and Applied Algebra, 102:251–272, 1995.<BR><BR> [BCF63] R.O. Burdick, P.E. Conner, and E.E. Floyd. Chain theories and their derived homology. Proceedings of the AMS, 19(5):1115–1118, Oct. 1963.<BR><BR> [Koc78] S. O. Kochman. A chain functor for bordism. Transactions of the American Mathematical Society, 239:167–196, 1978.<BR><BR></p> http://mathoverflow.net/questions/57656/standard-model-of-particle-physics-for-mathematicians/58252#58252 Comment by aleph0 aleph0 2011-10-31T20:01:24Z 2011-10-31T20:01:24Z These are not good books to learn from. I find these two to be to long and drawn out. They lack focus. They are missing and overall arc or plot and feel more like an amalgam of thousands of snippets written by different people with little regard to what the others were writing. These books may contain everything, but they also contain <i>everything</i>. On the plus side, I enjoy the historical annotations and stories. http://mathoverflow.net/questions/57656/standard-model-of-particle-physics-for-mathematicians Comment by aleph0 aleph0 2011-10-31T19:41:36Z 2011-10-31T19:41:36Z This is similar to this question at the Physics Exchange: <a href="http://theoreticalphysics.stackexchange.com/questions/222/quantum-field-theory-from-a-mathematical-point-of-view" rel="nofollow" title="quantum field theory from a mathematical point of view">theoreticalphysics.stackexchange.com/questions/&hellip;</a> http://mathoverflow.net/questions/51777/is-an-a-infinity-thing-the-same-the-same-as-strict-thing-viewed-through-a-homotpy Comment by aleph0 aleph0 2011-01-13T17:30:33Z 2011-01-13T17:30:33Z @Clark: I think that is indeed what I am asking. I am thinking of A_\infty as up-to-coherent-homotopy monoid/group/algebra/..., but I am trying to get a better feel of what exactly that means. It would be nice to be able to think that this was a strict (up-to-identity) monoid/... smudged by a homotopy equivalence. My question thus has two parts: (1) if I have a strict structure and smudge it through a homotopy equivalence, do I get an A_\infty structure? (2) If I have an A_\infty structure, can I assume it arose in this way?