2
$\begingroup$

I'm trying to understand why the category of presheaves on a small category $\mathbf C$, the functor category $[\mathbf C^{\mathrm{op}},\mathbf{Sets}]$, is an elementary topos. Right now I need to find a classifier, and to do so I'm reading Section 4 of the first chapter of Sheaves in Geometry and Logic, the one called Typical Subobject Classifiers.

Here they explain how to define a classifier introducing sieves, but I need to do so without using them. I have to use, for each object $C$ of the category $\mathbf C$, the subfunctors of $\mathrm{Hom}_{\mathbf C}(-,C)$ instead of sieves on $C$. Now I look for a classifier $1\stackrel{\mathrm{true}}\longrightarrow\Omega$.

The functor $$1\colon \mathbf C^{\mathrm{op}}\rightarrow\mathbf{Sets},\qquad C\mapsto \{*\}$$ is a terminal object for $[\mathbf C^{\mathrm{op}},\mathbf{Sets}]$.

I define the functor $\Omega\colon \mathbf C^{\mathrm{op}}\rightarrow\mathbf{Sets}$ like this: for each object $C$ of $\mathbf C$, let $$\Omega(C)=\mathrm{Sub}_{[\mathbf C^{\mathrm{op}},\mathbf{Sets}]}\big(\mathrm{Hom}_{\mathbf C}(-,C)\big)$$ be the set of subfunctors of $\mathrm{Hom}_{\mathbf C}(-,C)$; and for each arrow $C'\stackrel f\to C$, let $$\Omega(f):\Omega(C)\rightarrow\Omega(C'),\qquad P\mapsto\Omega(f)(P),$$ where $\Omega(f)(P)$ makes the diagram $$\require{AMScd} \begin{CD} \Omega(f)(P) @>>> P \\ @V V V @VVV\\ \mathrm{Hom}_\mathbf C(-,C') @>>{f\circ-}> \mathrm{Hom}_\mathbf C(-,C) \end{CD} $$ into a pullback.

I define the natural transformation $1\stackrel{\mathrm{true}}\longrightarrow\Omega$ by $$\mathrm{true}_C\colon 1(C)=\{*\}\longrightarrow\Omega(C),\qquad *\mapsto\mathrm{Hom}_\mathbf C(-,C)$$ for all $C$ in $\mathbf C$.

And now, if I'm not wrong, I hope to prove that $1\stackrel{\mathrm{true}}\longrightarrow\Omega$ is a classifier for $[\mathbf C^{\mathrm{op}}, \mathbf{Sets}]$. I take a monic $Q\to P$ in $[\mathbf C^{\mathrm{op}}, \mathbf{Sets}]$ and I need to prove that there exists a unique natural transformation $\varphi$ that forms a pullback like this one: $$\require{AMScd} \begin{CD} Q @>>> 1 \\ @V V V @VV\mathrm{true}V\\ P @>>\varphi> \Omega \end{CD} $$

For every object $C$ of $[\mathbf C^{\mathrm{op}}, \mathbf{Sets}]$ and every element $x\in PC$, i know that this $\varphi$ is a natural transformation such that $\varphi_C(x)$ is a certain subfunctor of $\mathrm{Hom}_\mathbf C(-,C)$, and I think that, for every object $B$ in $\mathbf C$, this subfunctor gives $$\big(\varphi_C(x)\big)(B)=\{g\mid g\colon B\to C,\ (Pg)(x)\in QB\}.$$

Now I need to understand why this $\varphi$ is the only one which makes the previous diagram into a pullback. I cannot follow the proof of the book at this point. At page 39 they take a random natural transformation $\theta$ that makes the diagram into a pullback, and they prove that it needs to be equal to $\varphi$.

Given $x\in PC$ and an arrow $f\colon A\to C$, I can understand that $(Pf)(x)\in QA$ if and only if $$(\Omega f)(\theta_A x)=\text{true}_A(*),$$ and this means that the diagram $$\require{AMScd} \begin{CD} \mathrm{Hom}_{\mathbf C}(-,A) @>>> \theta_Cx \\ @V V V @VVV\\ \mathrm{Hom}_\mathbf C(-,A) @>>{f\circ-}> \mathrm{Hom}_\mathbf C(-,C) \end{CD} $$ is a pullback. But, from this point, how can I deduce that $\theta=\varphi$?

$\endgroup$
  • 1
    $\begingroup$ It's been a while since I've looked at this stuff, but aren't sieves on $C$ in one-to-one correspondence with subfuctors $F \subseteq \hom(-,C)$? (with $F(X)$ being the arrows in the sieve with domain $X$) $\endgroup$ – Hurkyl Sep 22 '16 at 20:31
  • $\begingroup$ Yes, that's what they wrote in the book. They say that it is "customary and useful" to forget subfunctors and talk about sieves instead. Why? It looks like they just do something to make this stuff easier and to convince everyone they're right. First of all: I'm still not sure whether a subfunctor is a subobject (that is, an equivalence class of monos) or a mono. They choose to ignore this "equivalence class" problem, and I'm getting mad trying to understand all of this as clearly as possible. $\endgroup$ – Miles Eagle Sep 23 '16 at 20:10
  • 1
    $\begingroup$ There's actually a third option: that $\subseteq$ refers to the set-theoretic "is a subset of" relation. That is, for functors with codomain in a ZFC-style category of sets, $F \subseteq G$ can mean that $F(X) \subseteq G(X)$ and $F(X \xrightarrow{f} Y) = G(X \xrightarrow{f} Y)|_{F(X)}$. I vaguely recall having the impression that this was what MacLane means by subfunctor. (although one should be able to replace subfunctors with monics) $\endgroup$ – Hurkyl Sep 23 '16 at 21:03
  • $\begingroup$ You remember correctly. The definition of subfunctor he gives is the same as this. End after this definition, he says, using your notation, that the inclusion $F\to G$ is a monic arrow in $[\mathbf C^\mathrm{op},\mathbf{Sets}]$, so that each subfunctor is a subobject. And then, he says that conversely, all subobjects are given by subfunctors To prove this, he takes a monic natural transformation $\theta:R\to P$, he defines $QC$ as the image of each injection $RC\to PC$ and then he says that $Q$ is a subfunctor of $P$, and $R$ is equivalent as a subobject to $Q$. $\endgroup$ – Miles Eagle Sep 23 '16 at 21:31
  • $\begingroup$ I cannot understand what a subfunctor is. Given a functor $G$, a subfunctor $F$ of $G$ is just what you wrote, another functor such that..., or is there something more? Is it a monic $F\stackrel\alpha\rightarrow G$? Is it the equivalence class of monics equivalent to $\alpha$? $\endgroup$ – Miles Eagle Sep 23 '16 at 21:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.