Equalizer and Coequalizer

In my algebraic topology lecture, I learned that product topology constructions are actually categorial products in Top and that, dually, coproducts are disjoint union spaces. We even learned that adjunction spaces are pushouts. But when talking about subspace and quotient space topology constructions, apart from the notions being dual to each other, no words about the underlying categorial concepts were lost.

Of course, these are just equalizers and coequalizers. While I found it rather obvious in the case of products & coproducts, I didn’t immediately see the correspondence for subspace and quotient space topologies. Specifically, given for example a subspace topology construction, how do you construct a diagram of which the subspace topology is an equalizer of?

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In my little article (my first one!) I try to answer that question. I start with defining the concepts of equalizers and coequalizers and show that every equalizer morphism is already a monomorphism. Duality then gives us that coequalizer morphisms are epimorphisms for free. Next I consider equalizers and coequalizers in Set. I give the concrete constructions, and show that in Set, conversly, every monomorphism may already be interpreted as an equalizer. The dual statement also holds, here the construction felt less obvious and less natural, where representatives of sets of pre-images need to be chosen and one is required to accept the axiom of choice.

With these results I show in the last section that subspace topology constructions do indeed correspond to equalizers and quotient space constructions to coequalizers.

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3.25 diagrams per page (26 total)