[000B] Definition 2.1 (Construction of the object of monos).

A mono from $X$ to $Y$ is characterized by the fact that its kernel pair is the diagonal of $X$.

The characteristic function of the diagonal $\Delta\Sub{X}$, written $\gl{=\Sub{X}}$, has the arrow ${=\Sub{X}} : \ObjTerm{\ECat}\to \Omega\Sup{X\times X}$ for its cartesian adjoint.

The “kernel pair” operation from $Y\Sup{X}$ to $\Omega\Sup{X\times X}$ (written $\hat{\phi}$) can be obtained as the adjoint of the arrow from $Y\Sup{X}\times X\times X$ to $\Omega$ that classifies the subobject $k:K\rightarrowtail Y\Sup{X}\times X\times X$ equalizing the following two horizontal maps:

(that is to say, in $\SET$, the set of triples $\prn{f,x\Sub{1},x\Sub{2}}$ such that $f\prn{x\Sub{1}} = f\Sub{x\Sub{2}}$).

The object $\ObjMono{X}{Y}$ is then defined by the following fiber product:

[000E] Translator’s Note 2.1·a.

We have translated Bénabou’s “équivalence nucléaire” as “kernel pair”. We have translated Bénabou’s “noyau du couple” as “equalizer” — a usage of the French school that is explained by Grothendieck in his 1965-1966 lectures on basic category theory (Introduction Au Langage Fonctoriel, 1966, I.6.3.2). I have also renamed some of the labels in the first commutative diagram of Definition 2.1 [000B] for clarity.