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Mathoverflow.net news digest

  • 0 days

    When is a classifying object the same as an object with vanishing higher homotop...

    Let $T$ be a topos over a site $S$. This topos comes with a cohomology theory $H^{\bullet}$ and the notion of homotopy groups $\pi_{\bullet}$.
    Assume I have an object $X\in S$, and I want to calculate its cohomology.
    If for instance the underlying site is the site $\operatorname{Top}_{*}$ of pointed topological spaces, the two definitions of "classifying" objects of a group agree:...

  • 0 days

    Witten confusion: supersymmetry and ... huh?

    I've been working through the derivations and use of equations (13) and (15) of Witten's `Supersymmetry and the Morse Inequalities' [https://projecteuclid.org/journals/journal-of-differential-geometry/volume-17/issue-4/Supersymmetry-and-Morse-theory...

  • 0 days

    Combinatorial explanation of webs / cubic planar maps formula

    Define a Kreweras web to be a simple (no multiple edges, no loops) plane graph drawn in a disc, with $3n$ labeled vertices on the boundary of the disc and some number of internal vertices, such that:
    all boundary vertices have degree 1, and all internal vertices have degree 3;...

  • 0 days

    Lifting functors from representables to representing objects

    Let $C$ and $D$ be two categories. The Yoneda lemma says that for eachobject $Y\in D$ and each functor $F\colon D\to \mathrm{Sets}$, we have abijection
    $$\mathrm{Nat}(h_Y ,F)\ \longrightarrow\ F(Y), \qquad w\ \longmapsto\ w_Y(1_Y),$$
    and in particular we have a fully faithful functor...

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