Paul VanKoughnett
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Registered User
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I'm a first-year grad student at Northwestern. I like homotopy theory and that thing where you stand across the street from people in a parka and yell at them.
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May 19 |
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Composition in the category quotient $g$ is a map $N'' \to P/P'$; taking a further quotient induces $N'' \to P/P''$; and then by definition, $N' \cap N''$ is a subobject of the kernel of this map, so it passes to $N''/(N'\cap N'')\to P/P''$. There's a good reason why arguments about abelian categories look like arguments about modules: the Freyd-Mitchell embedding theorem, which says that any small abelian category embeds into some category of modules over a ring. So any statement you can phrase in terms of an arbitrary small set of objects in an abelian category, and prove when those objects are modules, is true in general! |
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May 19 |
accepted | Composition in the category quotient |
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May 18 |
answered | Composition in the category quotient |
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May 16 |
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Computation of Ext(Z^N,Z) This should answer your question: mathoverflow.net/questions/40499/… |
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May 6 |
answered | Definition of subobject classifier in presheaves |
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May 3 |
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Split and pure exact sequence of sheaves Tensor products commute with direct sums, and if the sequence is split, $B \cong A \oplus C$. |
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Apr 28 |
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Is there a local projective model structure on simplicial sheaves? What are its fibrant objects? This paper (math.uiuc.edu/K-theory/0462/combination2.pdf) by Benjamin Blander seems to imply that the answer to your first question is 'yes,' and you again get a Quillen equivalent model category, with sheafifying and forgetting being a Quillen equivalence. |
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Mar 22 |
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Intuition as to why the K-theory of a ring should be the homotopy theory of an H-space That isn't really an answer to your questions, so for now let me just make an insinuation: what could you mean by a space 'naturally' having a multiplication on its homotopy groups, other than this being induced by a multiplication on the space itself? |
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Mar 22 |
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Intuition as to why the K-theory of a ring should be the homotopy theory of an H-space In fact the $K$-groups of a ring are naturally modelled as the homotopy groups of a spectrum. (If you want to stick with spaces, you can in turn model this as an infinite loop space, i.e. a space which can be expressed as the form $\Omega^nX_n$ for some space $X_n$, for every $n$.) When you're starting with a ring, the spectrum you get is always a ring spectrum, which is the spectrum version of an $H$-space. I don't actually know why this is true, but I'd guess it pops out of one of the various constructions of this spectrum. |
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Mar 5 |
accepted | injectivity is a local property over noetherian rings |
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Mar 4 |
awarded | ● Teacher |
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Mar 4 |
answered | injectivity is a local property over noetherian rings |
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Mar 4 |
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$\pi$-cohomology class — a variant of cohomology class In order for this to be well-defined, you'd need to say precisely what it means for two cycles to '"deform" into each other continuously.' If both cycles are subcomplexes of $X$, you could ask for them to be simplicially homotopic, but then you'd want your simplicial set to be Kan, i. e. probably the singular complex of the space $X$ rather than a triangulation. Then you'd need to make sure that your definition extends well to the abelian group of cycles. |
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Feb 27 |
awarded | ● Critic |
