Equaliser

An equaliser of a pair of morphisms $f,g:{{X}\to{Y}}$ in a Category $\mathcal{C}$ is an object $E:\mathcal{C}$ equipped with a morphism $e:{{E}\to{X}}$ such that

• $f\circ e=g\circ e$
• For any other $a:A\to X$ with $f\circ a=g\circ a$, there exists a unique universal map $u:A\to E$ with $e\circ u=a$.

Note that the first condition ensures that $E$ is a Bundle over $Y$ with $f\circ e=g\circ e$, though this is a bit trickier to think about from the bundle POV; the morphism $e$ acts as a bundle map to both $f:X\to Y$ and $g:X\to Y$, so it's somehow a "multimap"?

More generally, if Pullbacks are Cartesian Morphisms in the Codomain Fibration, then what are equalisers? Typically, this is handled by Equality in a Cartesian Fibration.

Properties

• Equalisers are Limits over the Walking Parallel Pair.
• Equalisers are Connected Limits