## Equalizers and Coequalizers (Pt. II)

**Point of Post: **This is a continuation of this post.

Now that we have defined the notion of equalizers we can dualize everything to define coequalizers. Namely, let be parallel arrows in . We define a pair where to be a *coequalizer *of and if and whenever also satisfies this property then there exists a unique arrow such that .

Let’s take a look at some examples:

If is a ring and is a pair of parallel arrows in then a coequalizer for and is given by where is the projection map .

If is a pair of parallel arrows in then a coequalizer of these arrows is where if there exists such that and , along with the natural projection .

As in the case of equalizers we shall generally denote a coequalizer (as an object) by or just .

Of course, we can apply the same logic as for the case of equalizer to conclude that:

**Theorem: ***Let be a category containing parallel arrows . Then, a pair is a coequalizer of and if and only if there exists a natural isomorphism with .*

Of course then we have that:

**Theorem: ***Coequalizers are unique up to isomorphism.*

Unsurprisingly, equalizers and coequalizers have to be monic and epic. Indeed:

**Theorem: ***Let be an equalizer for some pair of parallel arrows and , then is monic. If is a coequalizer for and then is epic.*

**Proof: **Suppose that is not monic, then there exists such that . We see then that . Thus, “equalizes” and and thus there exists a unique arrow such that . Clearly though, by assumption, can be taken to be either or and thus by uniqueness .

Dualizing this proof works for coequalizers.

**Kernels and Cokernels**

Suppose now that we are in an preadditive category . If is any arrow we define a *kernel *of to be an equalizer of and the zero arrow . Since kernels, being equalizers, are unique up to isomorphsim we often times don’t distinguish between different kernels and just write for a kernel of . Similarly, we define a *cokernel *for to be a coequalizer of and and denote this . What’s interesting though is that we could have defined kernel and cokernel for any category with a zero object, what makes things so interesting for preadditive categories is that equalizers and coequalizers are actually subsumed in kernels in cokernels for preadditive categories. Indeed, it’s fairly trivial to see that:

**Theorem: ***Let be a preadditive category and a pair of parallel arrows in . Then, and is an equalizer and coequalizer of and respectively.*

What’s kind of cool is that some of our familiar notions of kernels and cokernels carry over the general case of preadditive categories. For example:

**Theorem: ***Let be a preadditive category. Then, is monic if and only if .*

**Proof: **Suppose first that is monic, and let be the usual map. Then, we have that and thus so that . Conversely, if we note that if then and since this implies that so that .

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**References:
**

[1] Mac, Lane Saunders. *Categories for the Working Mathematician*. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. *Abstract and Concrete Categories: the Joy of Cats*. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. *Categories and Modules with K-theory in View*. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. *Abelian Categories.* New York: Harper & Row, 1964. Print.

[5] Mitchell, Barry. *Theory of Categories.* New York: Academic, 1965. Print.

[6] Herrlich, Horst, and George E. Strecker. *Category Theory: An Introduction*. Lemgo: Heldermann, 2007. Print.

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