Name for a set of pairs of elements that equalise two functions?

Question

Is there an established name for this $eql$ function?

$$\operatorname{eql}(f, g) = \{\ (x, y)\mid f(x) = g(y)\ \}$$

In category theory, such a set is exactly the equalizer object of the arrows $f$ and $g$ in the category of sets. — MJD, May 17 '14 at 16:27
According to Wikipedia it's called an equalizer even outside of category theory. — MJD, May 17 '14 at 16:35

score 0 · Answer 1 · answered May 18 '14 at 03:34

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According to Wikipedia this set is called an equalizer of the two functions.

answered May 18 '14 at 03:34

MJD

score 0 · Accepted Answer · answered May 19 '14 at 10:41

I've discovered through Reddit (see Bill's link) that this is called the pullback:

$$pb(f, g) = \{\ (x, y) \mid f(x) = g(y)\ \}$$

The equalizer is:

$$eql(f, g) = \{\ x \mid f(x) = g(x)\ \}$$

But we can define each in terms of the other:

$$pb(f, g) = eql(f \circ \pi_1, g \circ \pi_2)$$

$$eql(f, g) = pb(f, g) \cap pb(id, id)$$

2 Answers2