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Does the singular functor $sing: Top\rightarrow SimpSet$ form the category of topological spaces to simplicial sets commutes with reflexive coequalizers?

Recall that the singular functor $sing$ is the right adjoint to the geometric realization functor $|-|:SimpSet\rightarrow Top$.

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No; in fact it doesn't even send quotients of equivalence relations between CW complexes to epimorphisms. The simplest example is the usual quotient map from the interval to the circle: it doesn't induce a surjectivity on maps out of the interval, so it doesn't induce an epimorphism on singular simplicial sets.

It would be pretty strange if Sing did preserve reflexive coequalizers, since it preserves coproducts: this would imply it preserves all colimits, even though it has no right adjoint.

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