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Monday, November 6, 2017

**Abstract:** I will discuss the paper $``$A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets$''$ by Alessandro Achille and Alessandro Berarducci $[\href{https://arxiv.org/abs/1706.02094}{arXiv}]$. In this paper, they compare the definable homotopy groups of a definable space $X$ (denoted $\pi_n(X)^\mathrm{def}$) with the (usual) homotopy groups of the quotient $X/E$, where $E$ is a certain kind of type-definable equivalence relation on $X$. Under certain assumptions, these groups are actually isomorphic. These types of quotients were first studied by Pillay in the case that $X$ is actually a definable group. No knowledge of model theory will be assumed. In particular ``definable,'' ``type-definable'' and ``o-minimal'' will all be defined.