# Equalizer and Coequalizer

**Posted:**November 18, 2011

**Filed under:**category theory Leave a comment

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?

*Download the pdf.*

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.

Download the pdf and the source code.

*3.25 diagrams per page (26 total)*