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Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the compo …

Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of topological spaces on $X$. Then, the map $U\mapsto \pi_1(\mathcal{F}(U))$ for $U\subseteq X$ open is a gerbe over $X$. I learned this e …

Let $M,N$be two smooth manifolds. Let $f,g:M\rightarrow N$ be two smooth maps. We have the notion of a homotopy (smooth homotopy) from the maps $f$ to the map $g$. Is there a similar concept for morp …

There is a notion of $K$-theory for a manifold $M$. Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is …

I am looking for some examples of failure of some structures giving interesting notions. For example, we have the following situation: Let $P(M,G)$ be a principal bundle. Let $\Gamma\subseteq TP$ be …

Some constructions can be seen as objects representing a functor. For example, Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \ma …

As mentioned by the user SGP, the book Supersymmetry and Equivariant de Rham Theory by Victor W Guillemin and Shlomo Sternberg discuss about Cartan model. One of the intentions is to prepare the read …

Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi_1(X)$. Here, the fundamental groupoid $\Pi_1(X)$ is made into a topological groupoid giving a topology on the morph …

I think what you need is a book on Homological algebra that discusses some category theory, some homology and group cohomology. You can try A Course in Homological algebra by Peter Hilton and Ur …

How does one introduce, or how were you introduced to characteristic classes? You can assume that the student is comfortable with principal bundles and connections on principal bundles. I am not as …