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I will answer the new version of the question: Does $\ker(\varphi_{*,a})=\ker (\phi_{*,a})$ for all $a\in P$, where $\phi_{*,a}:T_aP\to T_{\phi(a)}X_0$ and $\varphi_{*,a}:T_aP\to T_{\varphi(a)}Y_0$ a …

There are many connections between Lie groupoids and homotopical algebra. In recent years, a particularly prominent connection is to the theory of simplicial presheaves, originally developed in the c …

Stacks form an (∞,1)-category. The latter informal notion has many equivalent implementations: simplicial category, topological category, quasicategory (also known as ∞-category), Segal category, com …

Contrary to what is claimed in the comments, I would argue that the definition given in nLab's Idea section is rigorous enough to be an actual definition in a research-level paper, possibly with an ad …

Pullbacks of stacks coming from Lie groupoids are not always equivalent to Lie groupoids. Take $G=H=\mathbb{R}$. Define $F(x)=0$ if $x\leq 0$ and $F(x)=exp(−1/x^2)$ if $x>0$. The pullback is not eq …

There are analogues of Lie's theorems in homotopy theory, primarily for rational and $p$-adic homotopy types, as well as Lie ∞-groupoids, which can be seen as smooth homotopy types. In the rational ca …

There is no introductory book on Lie ∞-groupoids and ∞-algebroids analogous to Mackenzie's book. The only book-length treatment that covers these subjects is Urs Schreiber's Differential cohomology in …

That is, what algebraic structure captures this kind of groupoid-with-restriction and how do we describe its action on a given sheaf more precisely? This structure is well known and has many equival …

If C is a cartesian closed category with finite limits, then so is the category of internal groupoids in C. Indeed, the internal hom can be constructed by replicating the usual definitions of a functo …