Definition 24.3.1. A topologically enriched category is a classical 1-category \(D\) together with a chosen topology on each of the hom sets \(\Hom _D(X,Y)\) for \(X,Y \in D\), satisfying the property that the composition maps \(\Hom _D(Y,Z) \times \Hom _D(X,Y) \to \Hom _D(X,Z)\) are continuous with respect to these topologies. We denote by \(\Homtop _D(X,Y) \in \Top \) the hom set in \(C\) equipped with the chosen topology.
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