Sieve


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Sieve

Definition 1.12   A sieve on an object $ A\in\mathscr{C}$ is a collection S of morphisms in $ \mathscr{C}$ whose codomain is A and such that, if $ f:B\rightarrow A\in S$ then, given any morphisms $ g:C\rightarrow B$ we have $ f o g\in S$, i.e. S is closed under left composition:

$\displaystyle \xymatrix{ B\ar[rr]^{f}&&A\\ &&\\ C\ar[uu]^{g}\ar[rruu]_{fog}&&\\ }$

For example in a poset a sieve is an upper set. Specifically, given a poset C, a sieve on $ p\in C$ is any subset S of C such that if $ r\in S$ then 1) $ p\leq r$ 2) $ r^\lq \in S\hspace{.1in}\forall r\leq r^\lq $.
A map $ \Omega_{qp}:\Omega_p\rightarrow\Omega_q$ between sieves exists iff $ p\leq q$ and given $ S\in\Omega_q$ then

$\displaystyle \xymatrix{ A\ar[rr]^{i_A}\ar[rrdd]_f&&A+B\ar[dd]^{[f,g]}&&B\ar[ll]_{i_B}\ar[ddll]^g\\ &&&&\\ &&C&&\\ }$

$\displaystyle \Omega_{qp}(S):=\uparrow p\cap S$    

where $ \uparrow p:=\{r\in C\vert p\leq r\}$

An important property of sieves is the following: if $ f:B\rightarrow A$ belongs to S which is a sieve on A, then the pullback of S by f determines a principal sieve on B, i.e.

$\displaystyle f^*(S):=\{h:C\rightarrow B\vert f o h \in S\}=\{h:C\rightarrow B\}=\downarrow B$    

$\displaystyle \xymatrix{ &&C\ar[dl]\\ &B\ar[dl]&\\ A&&D\ar[ll]\\ }\hspace{.2in} \xymatrix{ &&\\ \ar@{=>}[rr]^{f^*}&&}\hspace{.2in} \xymatrix{ &&\\ B\ar@(ul,ur)[]^{i_B}&&C\ar[ll] }$

The principal sieve of an object A, denoted by $ \downarrow A$, is the sieve that contains the identity morphism of A therefore it is the biggest sieve on A.
An important properties of sieves is that the set of sieves defined on an object forms an Heyting algebra (see 3.5), with partial ordering given by subset inclusion.


Cecilia Flori 2007-01-02