2. The object of monos [000A]
Let $X$ and $Y$ be two objects of a topos $\ECat$. We propose to define a subobject of $Y\Sup{X}$ whose points correspond exactly to monomorphism from $X$ to $Y$.
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:
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.
2.2. Universal property of the object of monos [000C]
For each object $T$ of the topos $\ECat$, we have a natural bijection $\Hom{\ECat}{T}{\ObjMono{X}{Y}}\cong \Monos{\Sl{\ECat}{T}}{X\times T}{Y\times T}$ (the second term representing the set of monomorhpisms from $X\times T$ to $Y\times T$ in $\Sl{\ECat}{T}$).
Proof. Via the cartesian adjunction, the arrows $g:T\to Y\Sup{X}$ correspond bijectively to arrows $\hat{g} : T\times X \to Y$; the latter are in bijection with arrows $g\tick = \prn{\hat{g},p} : T\times X \to T\times Y$, where $p$ denotes the projection $T\times X \to T$.
We also show that the condition “$\hat{\phi}$ factors through $=\Sub{X}$” is equivalent to “$g\tick$ is a mono”. But the first amounts to the commutativity of the inner rectangle below:
The annotation $\text{(f.p.)}$ follows Bénabou’s $\text{(p.f.)}$ in the diagram in the proof of Proposition 2.2·a [000D] and denotes a fibered product square, i.e. a cartesian square.
If the rectangle commutes, the composite $\prn{g\times X\times X}\circ\prn{T\times \Delta}$ factors through $K$ as $k\circ\bar{g}$ and the square $(1)$ is a fiber product (taking account of the fact that $T\times\Delta$ is a mono); likewise, if the fibered product of $k$ and $g\times X\times X$ is a square of type $(1)$, then the inner rectangle is commutative (by associativity of fibered products).
The condition on $(1)$ will turn out to be satisfied if and only if $T\times \Delta$ is the equalizer of the composites $\Con{ev}\circ \prn{T\times\pi\Sub{1}}$ and $\Con{ev}\circ \prn{T\times\pi_2}$ (according to a classical lemma) the composites $\hat{g}\circ \prn{T\times\pi\Sub{1}}$ and $\hat{g}\circ\prn{T\times\pi\Sub{2}}$.
It remains to show that this property is equivalent to the fact that $\prn{\hat{g},p}$ is a mono, which can be verified in terms of sets: if we denote by $\prn{t,x\Sub{1},x\Sub{2}}$ an arbitrary element of $T\times X\times X$, the two properties amount to the condition “$\hat{g}\prn{t,x\Sub{1}} = \hat{g}\prn{t,x\Sub{2}}$ entails $x\Sub{1}=x\Sub{2}$”. ∎
If $T=\ObjTerm{\ECat}$, we obtain a natural bijection
Thus the points of $\ObjMono{X}{Y}$ correspond exactly to the monos from $X$ to $Y$ in $\ECat$. But these do not entirely determine the object $\ObjMono{X}{Y}$.
2.3. Composition of monos [000F]
If $X$, $Y$, $Z$ are objects of a topos $\ECat$, there exists a unique arrow from $\ObjMono{X}{Y}\times\ObjMono{Y}{Z}$ to $\ObjMono{X}{Z}$ factorizing the composition $\Con{C}$ from $Z\Sup{Y}\times Y\Sup{X}$ to $Z\Sup{X}$.
Proof. TODO
TODO
Bibliography
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Grothendieck, A. (1966). Introduction au Langage Fonctoriel. https://agrothendieck.github.io/divers/ilfg.pdfDetails