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). It is defined by iterating the construction

F^+(X) = \varinjlim\sb {R} \varprojlim\sb {Y \in R} F(Y) http://latex.mathoverflow.net/png?F%5E%2B%28X%29%20%3D%20%5Cvarinjlim%5F%7BR%7D%20%5Cvarprojlim%5F%7BY%20%5Cin%20R%7D%20F%28Y%29

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.)

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). 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 $X$. 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.)

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) http://latex.mathoverflow.net/png?F%5E%2B%28X%29%20%3D%20%5Cvarinjlim%5F%7BR%7D%20%5Cvarprojlim%5F%7BY%20%5Cin%20R%7D%20F%28Y%29

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.)

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). It is defined by iterating the construction

F^+(X) = \varinjlim\sb {R} \varprojlim\sb {Y \in R} F(Y) http://latex.mathoverflow.net/png?F%5E%2B%28X%29%20%3D%20%5Cvarinjlim%5F%7BR%7D%20%5Cvarprojlim%5F%7BY%20%5Cin%20R%7D%20F%28Y%29

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.)