nLab cosieve

Cosieve is a dual notion to sieve; that is, a cosieve in $C$ in a sieve in the opposite category $C^{op}$. They can be used to define Grothendieck cotopologies, dual to Grothendieck topologies.

A cosieve $R$ under an object $x$ in $C$ is a family of morphisms $f:x\to y$ in $C$ with domain $x$ closed under postcomposition with any morphism in $C$. In other words, $f\in R$ implies $h\circ f\in R$ whenever the composite $h\circ f$ exists. Cosieves under $x$ are also said to be cosieve in the under category $x \backslash C$; all such cosieves for varying $x$ are said to be cosieves on $C$. Cosieves may be viewed as subfunctors of the (co)representable (covariant) functors $h^x=C(x,-)$.

Cosieves on $C$ may be organized into a category $\mathrm{coSv}(C)$. For convenience we will note the domain $x$ of a sieve as a part of the data. Thus objects of $\mathrm{coSv}(C)$ are pairs of the form $(x,R)$ where $x\in\mathrm{Ob}(C)$ and $R$ is a cosieve in $x \backslash C$. A morphism $(x,R)\to (x',R')$ is a map $f:x\to x'$ such that the cosieve $R'\circ f = \{g\circ f | g\in R'\}$ is a subset of $R$. The usual composition of underlying morphisms in $C$ defines a composition in $\mathrm{coSv}(C)$, because $R''\circ (g\circ f)= (R''\circ g)\circ f\subset R'\circ f\subset R$ where $g:(x',R')\to (x'',R'')$. Note that $coSv(C) = Sv(C^{op})^{op}$.