Coequalizer

Coequalizer

The coequalizer of two functions f and g, where f and g take elements of X to elements of Y, is defined as the quotient Y / ~, where ~ is the equivalence relation generated by observing

$$f(x) = g(x) for all x in X$$

It is easiest to understand the nature of coequalizers by considering two maps (x) => x defined as:

$$f(x) = x g(x) = x + 1$$

We must have that f(1) = g(1), so 1 ~ 2, 3.14 ~ 4.14, and so on. The coequalizer is the half open interval [0, 1). It is as if we have wrapped the real number line around in a helix with a period of 1 and equated everything that lies directly above or below one another. Classes arise by recognizing vertically aligned positions in the helix as being equivalent to one another. A circle falls out of projecting these "classes" down..

Alternatively, one could imagine drawing the line y = x and y = x + 1 on the same graph. The two parallel lines can be understood as follows. Look at f(3) = 3 and g(3) = 4. One can look directly "up" to g(3) from (3, f(3)). The same can be done for any (x, f(x)). Now, if we add ε to 3, we get f(3 + ε) = 3 + ε = 3 + ε + 1. Keep incrementally sliding the "up arrow" to the right on our graph until we evaluate f(4) = 4 and find that 4 ~ 5. This means 3 ~ 5 by transitivity, just as 3 + ε ~ 4 + ~. Rather than imagining coiling the real number line into a helix of period 1, we slid along our graph and noticed equivalence classes arise at sliding intervals of 1. We have the interval [0, 1) as a coequalizer. We could use [122, 123), but we choose [0, 1) for simplicity.