## Equalizers and Coequalizers (Pt. I)

**Point of Post: **In this post we discuss the notion of equalizers and coequalizers in general categories, as well as discussing the particular cases of kernels and cokernels in -categories.

*Motivation*

In this post we discuss another type of universal arrow which shall be very important for our soon-to-come endeavors–equalizers and coequalizers. To motivate why one would want to come up/care about equalizers and coequalizers let us engage in a thought experiment. As of now we have taken several normal ideas that we have encountered in “real mathematics” and attempted to abstract them to categorical definition. Continuing in this way, let’s see if we can abstract the notion of a kernel for maps between abelian groups. In other words, we know that given two abelian groups and and a group map we have great interest in the kernel, , of which concretely is defined as the preimage of under –how can we define the kernel of in a categorical way? This is a slightly non-trivial matter since we cannot make any specific reference to elements, only to maps. Well let’s see if we can work this out. The obvious way to replace as a set with a more categorical way of thinking about it, is to think of it instead as just some object with an inclusion along with certain properties. What properties, you ask? Well clearly we want and moreover we want to be the minimal such map for which this is true. In other words, if is any other abelian group and we have a map with the property that then and so we can factor through to get a diagram of the form

This formulation of kernel easily now generalizes to defining the kernel of a map in any category with a zero object. Indeed, suppose that is such a category and suppose we have a diagram in . Then, we define a kernel of to be an object along with an arrow with the property that and is universal with respect to this property. Note though that, from basic stuff, we have that . And so we see that and if is an arrow such that then factors through . Thus, we can also think of our kernel as being an “equalizing” arrow for and which is universal with respect to this property. This naturally leads us to the notion of equalizers. Dualizing all of this gives coequalizers, and specifically cokernels.

*Equalizers and Coequalizers*

We begin by abstracting the above notion of kernel to give the full-blown definition of an equalizer. Indeed, suppose that is some category with objects and and we have arrows . An *equalizer* of and is an ordered pair where is some object of and an arrow with the property that and whenever is another such arrow then there exists a unique arrow such that .

Let’s discuss some particular types of equalizers, get some good old examples under our belt.

Equalizers always exist in , indeed suppose that are two -maps, and consider the inclusion . Clearly then and if then so that so that factors through . Moreover, it’s clear that such a factoring is unique.

Equalizers also exist, always, in , in fact, most times the equalizers in a given concrete category shall just be the equalizer in endowed with the necessary structure. Indeed, suppose we have two set maps , we can then just define and prove that the inclusion gives an equalizer of and . Indeed, evidently and if then and so we can construct a factorization of through which is evidently unique.

In we can just endow the set theoretic equalizer of two arrows to give a topological equalizer.

To make our lives easier, we shall often denote the equalizer of two parallel arrows by or, if we are being extra lax on noation, .

We can interpret the definition of an equalizer of two arrows slightly differently. Namely, suppose that we have two parallel arrows . Consider the object map which takes to and which takes an arrow to the arrow given by to . To prove that this actually makes sense, we need to prove that if then . To do this we merely need to check that if then , but this is obvious since

What we now claim is that this pair of object and arrow maps fits together to form a (contravariant) functor, call it . But, this is really easy since we see that our functor is really nothing but the functor with the domain restricted–in particular, the two mapping properties of a functor have the same proof as they did for the contravariant Hom functor.

Ok, cool, so now we know that we have this functor , so how does this help define equalizers differently? Well, I claim that an object of is an equalizer for if and only if the functor is naturally isomorphic to the contravariant Hom functor . Indeed, suppose for a second that was such an object. What we have then is a natural isomorphism . But, Yoneda’s lemma tells us that we can associate to this an element defined by . But, of course we see that is then an arrow which equalizes and , on in other words, for which or . Now, suppose that is another arrow equalizing and . We see then that and so there exists a unique such that . We claim that . Indeed, writing down the naturality diagram for and the map gives us the following

Plugging in tells us that , or rewriting this says that or as desired. Thus, we really do see that if is an object such that the functor is isomorphic to then is an equalizer.

Conversely, it’s not hard to see that if is an equalizer for then there exists a natural isomorphism such that . Putting this all together we see that:

**Theorem: ***Let be a (locally small) category and a pair of parallel arrows in . Then, is an equalizer for and if and only if there exists a natural isomorphism such that .*

In particular, by Yoneda’s lemma we may conclude that:

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

**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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