show/hide this revision's text 3 converted images to tex

A presheaf F $F$ with values in C $C$ is a called a sheaf if, for every object X $X$ and every covering sieve R $R$ of X, $X$, the natural maps

$F(X) \rightarrow F(Y)$

for each Y in R induce an isomorphism

$F(X) \xrightarrow{\sim} \varprojlim_{Y \in R} F(Y)$

This definition makes sense without any assumptions on C.$C$.

The sheafification construction makes use of filtered colimits and essentially arbitrary limits (if you are interested in sheaves on a particular topology then you might be able to restrict the class of limits that need to be considered). It is defined by iterating the construction

$F^+(X) = \varinjlim_{R} \varprojlim_{Y \in R} F(Y)$

where the $\varinjlim$ is taken over covering sieves of X. $X$. If F $F$ is set-valued, the associated sheaf of F $F$ is $F^{++}$.

I don't know what conditions on C $C$ are necessary to make the sheafification of a presheaf in C $C$ a sheaf, but I wouldn't expect the construction to behave very well unless C $C$ is a fairly special category.

(Categories of algebraic structures on sets defined by finite inverse limits would qualify as "special", essentially because filtered colimits commute with finite inverse limits.)
show/hide this revision's text 2 forgot to mention initially that the + construction must be iterated

A presheaf F with values in C is a called a sheaf if, for every object X and every covering sieve R of X, the natural maps

$F(X) \rightarrow F(Y)$

for each Y in R induce an isomorphism

$F(X) \xrightarrow{\sim} \varprojlim_{Y \in R} F(Y)$

This definition makes sense without any assumptions on C.

The sheafification construction makes use of filtered colimits and essentially arbitrary limits (if you are interested in sheaves on a particular topology then you might be able to restrict the class of limits that need to be considered). The definition It is defined by iterating the construction

F^+(X) = \varinjlim\sb {R} \varprojlim\sb {Y \in R} F(Y)

where the $\varinjlim$ is taken over covering sieves of X. If F is set-valued, the associated sheaf of F is $F^{++}$.

I don't know what conditions on C are necessary to make the sheafification of a presheaf in C a sheaf, but I wouldn't expect the construction to behave very well unless C is a fairly special category.

(Categories of algebraic structures on sets defined by finite inverse limits would qualify as "special", essentially because filtered colimits commute with finite inverse limits.)
show/hide this revision's text 1

A presheaf F with values in C is a called a sheaf if, for every object X and every covering sieve R of X, the natural maps

$F(X) \rightarrow F(Y)$

for each Y in R induce an isomorphism

$F(X) \xrightarrow{\sim} \varprojlim_{Y \in R} F(Y)$

This definition makes sense without any assumptions on C.

The sheafification construction makes use of filtered colimits and essentially arbitrary limits (if you are interested in sheaves on a particular topology then you might be able to restrict the class of limits that need to be considered). The definition is

F^+(X) = \varinjlim\sb {R} \varprojlim\sb {Y \in R} F(Y)

where the $\varinjlim$ is taken over covering sieves of X.

I don't know what conditions on C are necessary to make the sheafification of a presheaf in C a sheaf, but I wouldn't expect the construction to behave very well unless C is a fairly special category.

(Categories of algebraic structures on sets defined by finite inverse limits would qualify as "special", essentially because filtered colimits commute with finite inverse limits.)