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In general, if $X_\bullet$ is a simplicial space, then its geometric realization is given by $$|X| = \left( \coprod_{n \ge 0} X_n \times \Delta^n \right) \biggm/ \sim,$$ where the equivalence relation is generated by $$(f^*x, u) \sim (x, f_*u),$$ where $f : [m] \to [n]$ is a morphism in the simplicial category $\Delta$ (ie. it's a nondecreasing map $\{0 < ...


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A topological category is an internal category in topological spaces, i.e. a space of objects $C_0$ and one of morphisms $C_1$ with maps $i:C_0\to C_1$ and $s,t:C_1\to C_0$ satisfying the axioms of a category. A topological category gives rise to a single space via a geometric realization that's inspired by the geometric realization of abstract simplicial ...


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Because the author introduced the normalization construction immediately prior to defining $\pi_n(V)=H_n(V)$, it looks like he is indeed thinking of the homology of the normalization. On the other hand, in the following sentence, he says that the homology of $V$ (with respect to the second differential) is the same as the homotopy, so my guess is that both ...



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