Franklin's Notes

Sieve

Given a category $\mathsf C$ and an object $x\in\mathsf C$, let $S_x$ denote the collection of all arrows of $\mathsf C$ with domain $x$. (This is also the collection of objects of the comma category $\mathsf C\uparrow x$.) Notice that $S_x$ "absorbs left-composition", i.e. for any $f\in S_x$ with $f:x\to y$ and $g:y\to z$ any other morphism of $\mathsf C$, we have that $gf:x\to z$ is also in $S_x$. A sieve on $x$, or an $x$-sieve, is a subcollection of $S_x$ that also absorbs left-composition. Alternatively, a sieve on $x$ can be thought of as a "left subideal" of $S_x$.

If $\mathsf C$ is a small category, we can define a functor $\Omega:\mathsf C\to\mathsf{Set}$ as follows. If $x\in\mathsf C$ is an object, then $\Omega(x)$ is defined as the set of $x$-sieves: whereas if $f:x\to y$ is an arrow of $\mathsf C$, then $\Omega(f):\Omega(a)\to\Omega(b)$ is defined by sending each $x$-sieve $S\in\Omega(a)$ to the $y$-sieve consisting of morphisms $g:y\to z$ such that $gf\in S$. It turns out that $\Omega$ is the codomain of the subobject classifier in the functor category $\mathsf{Set}^\mathsf C$, where $\top:\Delta_1\Rightarrow\Omega$ is the natural transformation with components $\top_a:1\to \Omega(x)$ sending $0\mapsto S_x$, the largest sieve on $x$. Thus, the truth values of a topos taking the form $\mathsf{Set}^\mathsf C$ can be determined by examining the sieves on $\mathsf C$.