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Let $X$ be a topological space, $\mathcal{C}$ be a locally small category with "good properties" (such as having small inverse limit, small filtrant inductive limit...etc.) and $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$.

The + functor turns $\mathcal{F}$ to another presheaf $F^{+}$ such that: For any open subest $U$ of $X$,

$$ F^{+}(U) = \varinjlim_{\mathcal{U}} \ \mathcal{F}(\mathcal{U}), $$ here $\mathcal{U}$ is a open convering of $U$ and $\mathcal{F}(\mathcal{U})$ is the equalizer of

$$ \prod_{V \in \mathcal{U}} \mathcal{F}(V) \rightrightarrows \prod_{V_1, V_2 \in \mathcal{U}} \mathcal{F} (V_1 \ \cap \ V_2) $$

One shows that $\mathcal{F}^{+}$ is a separated sheaf and if $\mathcal{F}$ is separated, $\mathcal{F}^{+}$ is a sheaf. So the sheafification is $\mathcal{F}^{++}$.

If $\mathcal{C}$ is a concrete category (everything are sets and maps), then the proof for these are clear.

My question is: For non-concrete categories, does one need some more conditions on $\mathcal{C}$ to prove these?

For example, in order to prove that $\mathcal{F}^{+}$ is separated, which is equivalent to

$$ \mathcal{F}^{+} (U) \rightarrow \mathcal{F}^{+} (\mathcal{U}) \mathrm{\ is \ a \ monomorphism}, $$

I found I need some conditions such as:

$$ \varinjlim_{i} \ \mathrm{Hom}(A, X_i) \rightarrow \mathrm{Hom}(A, Y) \mathrm{ \ is \ injective} $$ $$ \varinjlim_{i} \ \mathrm{Hom}(A, X_i) \rightarrow \mathrm{Hom}(A, \varinjlim_{i} \ X_i) \mathrm{ \ is \ surjective} $$

for some special filtrant inductive limits (those comes from open coverings of open subsets.)

The above conditioins are just from trying to mimic the proof when $\mathcal{C}$ is concrete. For example, for an element $x$ in a filtrant inductive limit, one can find some index $i$ and an element $x_i \in X_i$ such that $x$ is the image of $x_i$.

I would like to know if such conditions are really necessary. If not, how one proves the properties of $\mathcal{F}^{+}$?

Edit: This question have been downvoted once, and I would like to know why. Is this question is not suitable for mathoverflow or it is just not the question of someone's favor?

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This probably says more about my biases, but in my view there are only set-valued sheaves; everything else comes from lifting constructions on $\textbf{Set}$ to constructions on $\textbf{Sh}(X)$ (or more generally, a topos). Often these constructions coincide with the naive notion of a sheaf taking values in some other category, but there are also well-known examples where this fails: for instance, the notion of a local ring does not lift in this way. The point is that good constructions are compatible with the inverse image part of geometric morphisms, and sheafification is such a functor. – Zhen Lin Mar 21 '13 at 15:44
@Zhen Lin, I don't understand what this menas "lifting constructions on Set to constructions on Sh(X)". Also could you explain or give a reference that illustrate what is the relation between this with local ring, thank you. – user565739 Mar 22 '13 at 8:26
If you have a first-order logical theory $\mathbb{T}$ axiomatised in the so-called "geometric" fragment, then $\mathbb{T}$ can be interpreted in any sheaf topos, and inverse image functors will preserve models of $\mathbb{T}$. Any algebraic theory (e.g. groups, rings, modules, etc.) is geometric, but special properties of these theories mean they agree with the naïve construction via model-valued sheaves. Examples of theories that are not algebraic are: integral domains, local rings, fields, ... [continued] – Zhen Lin Mar 22 '13 at 8:32
In general, for a topological space $X$, a sheaf is a model of $\mathbb{T}$ precisely if every stalk is a model of $\mathbb{T}$. Thus, this approach of lifting constructions from $\textbf{Set}$ to $\textbf{Sh}(X)$ produces the right definition of "sheaf of local rings", which is a plus in my view. – Zhen Lin Mar 22 '13 at 8:33
@Zhen Lin, thank you for the comment, but if you could provide me some reference to these in order that I can get more precise ideas about these in the future. – user565739 Mar 23 '13 at 6:31
up vote 2 down vote accepted

You might be interested in the papers

P. Freyd, M. Kelly, Categories of continuous functors I, J. Pure Appl. Algebra 2 (1972), 1-18.

J.W. Gray, Sheaves with values in a category, Topology 3 (1965), 1-18.

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