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Let $G$ be a smooth connected linear algebraic group over an algebraically closed field (of characteristic zero if you wish). Let's consider (necessarily central) extensions of $G$ by the multiplicative group $\mathbb{G}_m$:$$1 \longrightarrow \mathbb...

On a smooth manifold $M$ with altas $\mathcal A=\{\phi:U_\phi\subseteq M\to \Bbb R^n\}_\phi$, we can define a tensor $T$ on $M$ as the collection $\{T_\phi\}_{\phi\in\cal A}$ such that whenever $U_\phi\cap U_\psi\neq\varnothing$ the transition rule ...

I am reading Strom's Modern Classical Homotopy Theory book.
In section 6.2.1 (page 147), cofibrant diagrams are discussed.
Briefly, let $T$ be the category of topological spaces and $I$ any (small) category. A functor $F : I \to T$ is called cofibrant if and only if for any $X, Y \in T^I$ and any $\theta : X \to Y$ and $\phi : F \to Y$ such that $\theta$ is a pointwise homotopy...

This is somewhat related to this old question of mine, but is hopefully easier.
Let $\mathcal{H}$ be the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric, and let $\mathcal{T}\subseteq\mathcal{H}$ be the subspace consisting of those $X$ which have $X=cl(int(X))$ and can tile the plane (i.e. there is...