MathOverflow

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:...

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...

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;...

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...