nLab
cosieve

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

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

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

Revised on July 11, 2009 05:22:07 by Toby Bartels (71.104.230.172)