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Given a concrete category C, with objects denoted Obj(C), and an equivalence relation ~ on Obj(C) given by morphisms in C. The moduli set for Obj(C) is the set of equivalence classes with respect to ~; denoted Iso(C). When Iso(C) is an object in the category Top, then the moduli set is called a moduli space.
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Differential of the evaluation map of the Kontsevich moduli space
Let $X$ be a smooth projective variety, and $\beta$ a curve class on it. We have the Kontsevich moduli space $\overline{\mathcal M}_g(X, \beta)$ of stable maps from genus $g$ curves to $X$ with class …