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A filtered colimit in a category is a colimit whose indexing category is a small filtered category. In the Category of sets, filtered colimits commute with finite limits in sets. A filtered colimit of modules is an example of this general categorical construction.

In a filtered colimit in a category of sets, any finite zigzag of generating identifications can be moved to one common stage. The filtered category conditions supply common objects and equalize the finitely many competing arrows. Consequently finitely many representatives and finitely many equalities can be realized simultaneously at one stage. This is the elementwise reason filtered colimits commute with finite limits in sets.

For a small category with finite limits, a set-valued functor preserving them has a category of elements with finite limits, so its opposite is a filtered category. The covariant density presentation is therefore a filtered colimit of representable functors. Conversely, representable functors preserve categorical limits, and filtered colimits commute with finite limits in sets, proving the criterion.

Every covariant representable functor $\mathcal{C}(c,-)$ preserves categorical limits, since a morphism out of $c$ into a limit is precisely a compatible cone. A filtered colimit of these functors is computed pointwise, and filtered colimits commute with finite limits in sets. Interchanging the finite limit and filtered colimit therefore proves that $F$ preserves finite limits, which is (i). This completes the cycle. It is also the flat functor and finite-limit preservation criterion for a finitely complete small category.

A filtered category is nonempty, every pair of objects maps to a common object, and every parallel pair becomes equal after postcomposition. Equivalently every finite diagram in a category has a cocone.

For a filtered set diagram $D:I\to \mathbf{Set}$, the finite-stage equality in a filtered set colimit says that representatives $x\in D(i)$ and $y\in D(j)$ define the same element exactly when their images agree at some common stage. Indeed, equality in the ordinary disjoint-union-and-quotient construction is witnessed by a finite zigzag; filteredness moves its objects to a common stage and equalizes the finitely many competing arrows.

For a finite categorical product, choose a common stage for all coordinate representatives to obtain surjectivity of the colimit-to-product comparison. Finitely many coordinate equalities can then be realized together at one later stage, proving injectivity. The empty product is preserved because a nonempty filtered category is connected. For an equalizer, an element whose two images agree in the colimit reaches a stage where they agree, so it is represented by an element of the stagewise equalizer; the same equality criterion proves injectivity. Finite products and equalizers construct every finite limit. Hence filtered colimits commute with finite limits in sets.

We prove the equivalence by the cycle solution, solution, solution, solution, then back to solution.

Suppose $F\cong {\mathrm{colim}}_{j\in J}{R}_j$, where $J$ is a filtered category and every $R_j$ is a representable functor. Every covariant representable functor $\mathcal{C}(A,-)$ preserves all existing categorical limits: maps from $A$ into a limiting object are exactly compatible families of maps into its diagram.

For a finite diagram in a category $K:I\to \mathcal{C}$, use pointwise colimits in the functor category, preservation by each $R_j$, and the permitted result that filtered colimits commute with finite limits in sets to obtainThe isomorphism is the canonical comparison, since every step respects the limiting projections. This also covers the empty finite diagram and hence the terminal object. Thus $(iv)\Rightarrow (i)$, completing the cycle:

The assertion that filtered colimits commute with finite limits in sets means that for every filtered category $\mathcal{J}$, every finite category $\mathcal{I}$, and every functor $D:\mathcal{I}\times \mathcal{J}\to \mathbf{Set}$, the canonical comparisonis a bijection. Surjectivity follows because an element on the right uses only finitely many representatives and compatibility equations, so filteredness moves all of them to one common stage. For injectivity, equality likewise consists of finitely many equalities, which become true at one later common stage.

The dual assertion fails in ${\mathbf{Set}}^{\mathrm{op}}$. Equivalently, cofiltered limits need not commute with finite colimits in sets. Index inverse systems by $n\in \mathbb{N}$, with arrows $n+1\to n$, and putusing the inclusions $A_{n+1}\hookrightarrow A_n$ and the unique maps $A_n\to B_n,C_n$. Since $A_n$ is nonempty, the pushout $B_n{\bigsqcup }_{A_n}{C}_n$ is a singleton at every stage, so its inverse limit is a singleton. Butand the pushout of the inverse limits is $1{\bigsqcup }_{\varnothing }1$, which has two elements. Thus a cofiltered limit fails to preserve this finite pushout.

Let $F$ be a presheaf of sets on a topological space $X$. The neighborhoods of $x$, ordered by reverse inclusion, form a filtered category, andThe first result therefore says that the stalk functor for presheaves of sets preserves finite limits.

For a set $S$, define a presheaf $R_{x}S$ bywith identity restrictions between neighborhoods of $x$ and the unique maps to $1$ otherwise. A natural transformation $F\to R_{x}S$ is exactly a compatible family of maps $F(U)\to S$ over neighborhoods of $x$, hence exactly a map $F_x\to S$. Thusso $R_x$ is right adjoint to the stalk functor.

Now restrict to sheaves. Suppose two morphisms $\alpha ,\beta :F\rightrightarrows G$ induce the same map on every stalk. For $s\in F(U)$ and every $x\in U$, equality of the two germs gives a neighborhood $V_x\subseteq U$ on which $\alpha (s)$ and $\beta (s)$ agree. The $V_x$ cover $U$, so the uniqueness axiom for $G$ gives $\alpha (s)=\beta (s)$. Therefore the joint stalk functor on sheaves of setsis faithful.

The presheaf $R_{x}S$ above is already a sheaf: an open containing $x$ has a cover member containing $x$, and compatibility forces one common element of $S$. Hence a right adjoint to $P$ sends a family $(S_x)$ to the product ${\prod }_x{R}_x{S}_x$, whose existence follows from the assumed closure of $\mathbf{Sh}(X)$ under limits.

Each stalk preserves finite limits, so $P$ preserves equalizers. If $P(f)$ is an isomorphism, its monicity and epicity are reflected by the faithful functor $P$; since $\mathbf{Sh}(X)$ is assumed balanced, $f$ is an isomorphism. Thus $P$ reflects isomorphisms. The Beck comonadicity theorem now applies: $P$ has a right adjoint, reflects isomorphisms, and preserves the required equalizers. Consequently $P$ is comonadic.

For presheaves of sets, $F_x={\underrightarrow{\lim }}_{U\ni x}F(U)$. This functor preserves finite limits because the neighborhood system is filtered and filtered colimits commute with finite limits in sets. Its right adjoint sends a set $S$ to the presheaf taking the value $S$ on opens containing $x$ and the singleton on other opens.