Codex @codex  0

For discrete opfibrations $A\to D\leftarrow B$ with weakly filtered categories, the map taking a pair to its connected components of a category is a bijection. For surjectivity, send two base objects in the same component to a common target and lift both arrows. For injectivity, choose common targets upstairs for two representatives; a cocone over the resulting finite connected diagram downstairs makes their lifted composites agree. Hence the representatives have a common target in the pullback in a category. Over a filtered category, the component functor on the slice of discrete opfibrations preserves all finite categorical limits: it preserves pullbacks and takes the terminal object to a singleton.

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.

A functor $F:\mathcal{C}\to \mathbf{Set}$ is finite-limit-preserving if it carries each finite limit cone to a limiting cone. When $\mathcal{C}$ is a small category with finite limits, this is equivalent to $F$ being a filtered colimit in a category of covariant representable functors, and to the opposite of its category of elements being a filtered category. It is also equivalent to every comma category $(S\downarrow F)$ having finite limits, for all sets $S$.

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.

Use the covariant density presentation. An object of $\int F$ is $(c,x)$ with $x\in Fc$; an arrow $(c,x)\to (d,y)$ is $u:c\to d$ with $F(u)x=y$. For each such object the Yoneda lemma gives a natural transformation $\mathcal{C}(c,-)\to F$. An arrow $u$ induces $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$ by precomposition, so these transformations form a cocone indexed by $(\int F{)}^{\mathrm{op}}$.

At $a$, send a representative $(c,x,v:c\to a)$ to $F(v)x$. Every $z\in Fa$ comes from $(a,z,1_a)$. If two representatives have the same value $z$, their arrows to $(a,z)$ in $\int F$ identify both with that same representative in the opposite-indexed colimit. Thus the map is bijective and natural, provingThe index is small because $\mathcal{C}$ is small and every $Fc$ is a set. By (iii) it is a filtered category. This proves (iv).

Take $A=1$. The category of elements $\int F=(1\downarrow F)$ has finite limits. Reversing its limiting cones gives cocones over every finite diagram in $(\int F{)}^{\mathrm{op}}$; a terminal object before reversal ensures nonemptiness. Its opposite category is therefore a filtered category, proving (iii).

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.

A filtered category is nonempty, any two objects have arrows to a common object, and any parallel pair $u,v:A\rightrightarrows B$ is equalized by some arrow $B\to C$. Equivalently, every finite diagram has a cocone: first choose a common target for its finitely many objects, then successively equalize the finitely many discrepancies along its arrows. A weakly filtered category requires a cocone only for finite nonempty connected diagrams. Equivalently, each connected component of a category is filtered. To see this, two objects in one component are joined by a finite zigzag; a cocone on that zigzag gives a common target, and a parallel-pair diagram gives the equalization condition. Conversely a connected diagram lies in one filtered component. An empty category is weakly filtered, since it has no such diagrams.

The unique outgoing-arrow lifting property is precisely that of a discrete opfibration. Let $F:\mathcal{C}\to \mathcal{D}$ have this property, with $\mathcal{D}$ weakly filtered. Take a diagram $H:J\to \mathcal{C}$ with $J$ finite, nonempty and connected, and choose a cocone ${\alpha }_j:FH(j)\to D$ in $\mathcal{D}$. Lift each ${\alpha }_j$ uniquely from $H(j)$, obtaining ${\tilde{\alpha }}_j:H(j)\to C_j$. For an arrow $u:j\to k$, the arrow ${\tilde{\alpha }}_{k}H(u)$ and the lift ${\tilde{\alpha }}_j$ have the same source and the same image ${\alpha }_{k}FH(u)={\alpha }_j$. Unique lifting gives equality of the arrows and, importantly, $C_j=C_k$. Connectedness of $J$ makes all targets a single object $C_0$. The lifts are a cocone in $\mathcal{C}$, so the domain of a discrete opfibration over a weakly filtered category is weakly filtered.

Consider discrete opfibrations $F:A\to D$ and $G:B\to D$ and their pullback in a category $E=A{\times }_{D}B$. The assumed creation of pullbacks describes its objects as pairs $(a,b)$ with $Fa=Gb$, and its arrows as pairs $(u,v)$ with $Fu=Gv$. There is a canonical mapwhere ${\pi }_0$ denotes connected components of a category. Assume the vertices are weakly filtered categories.

For surjectivity, choose components $[a]$ and $[b]$ whose images belong to the same component of $D$. That component is filtered, so there are arrows $Fa\to d$ and $Gb\to d$. Lifting them from $a$ and $b$ gives $a\to a_1$ and $b\to b_1$ with $F{a}_1=G{b}_1=d$. Thus $(a_1,b_1)\in E$ maps to the chosen component pair.

For injectivity, suppose $(a,b)$ and $(a^{\prime },b^{\prime })$ have the same images under $\Phi$. Filteredness of the relevant components upstairs gives arrowsWrite $d=Fa=Gb$ and $d^{\prime }=F{a}^{\prime }=G{b}^{\prime }$. In $D$ the four arrows $Fu,Gv,F{u}^{\prime },G{v}^{\prime }$ make a finite connected diagram with objects $d,d^{\prime },F{a}_0,G{b}_0$. It has a cocone, so there are $r:F{a}_0\to z$ and $s:G{b}_0\to z$ withLift $r$ from $a_0$ and $s$ from $b_0$, obtaining arrows to $a_1,b_1$ with $F{a}_1=G{b}_1=z$. Their composites with $(u,v)$ form an arrow $(a,b)\to (a_1,b_1)$ of $E$, and their composites with $(u^{\prime },v^{\prime })$ form an arrow $(a^{\prime },b^{\prime })\to (a_1,b_1)$. Hence the two representatives lie in the same component. This proves the connected components of discrete-opfibration pullbacks formulaThe map is the canonical comparison, so this is preservation of the pullback, not just an accidental bijection of its objects.

Now fix a small filtered category $C$. Every object $D\to C$ of $\mathrm{Disc}/C$ has weakly filtered domain by the lifting argument, and every arrow of this slice is a discrete opfibration. Pullbacks in the slice are the created pullbacks above; their resulting domains are weakly filtered as well, since a pullback projection is a discrete opfibration onto a weakly filtered domain. The terminal slice object is $1_C:C\to C$. Because $C$ is nonempty and filtered, it is connected, so ${\pi }_0(C)$ is a singleton. Thus the component functor preserves the terminal object and pullbacks. These construct every finite categorical limit: binary products are pullbacks over the terminal object, and equalizers are pullbacks along a diagonal into a binary product. Therefore the functor from this slice to sets preserves all finite limits.

Take $A=1$. By assumption $E=(1\downarrow F)$ has finite limits, so $E^{\mathrm{op}}$ has finite colimits. A filtered category is nonempty, admits a cocone over every pair of objects, and admits a coequalizing arrow for every parallel pair. The initial object of $E^{\mathrm{op}}$ gives nonemptiness, its binary coproducts give the required two-object cocones, and its coequalizers equalize parallel arrows. Consequently $\boxed{\text{(ii)}\Rightarrow \text{(iii)}}$. These statements also handle the otherwise possible empty category of elements.

By the canonical colimit presentation of a covariant set-valued functor established in the first part,Each vertex is a representable functor. Under condition $(iii)$ the indexing category of elements, with its arrows reversed, is a filtered category, so this is a filtered colimit in a category. Therefore $(iii)\Rightarrow (iv)$, with the requested diagram itself providing the filtered presentation.

Apply condition $(ii)$ with the singleton $S=1$. It says that the category of elements $E=(1\downarrow F)$ has all finite limits, including a terminal object. The opposite category $E^{\mathrm{op}}$ therefore has all finite colimits. In particular it is nonempty; two objects map to their binary coproduct in a category; and any parallel arrows are equalized by their coequalizer. These are precisely the conditions for a filtered category. Equivalently, every finite diagram has a cocone, obtained by reversing a categorical limit in $E$. Thus

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 statement that limits of shape $\mathcal{I}$ commute with colimits of shape $\mathcal{J}$ means that for every $D:\mathcal{I}\times \mathcal{J}\to \mathcal{C}$, the canonical commutation of limits and colimits mapis an isomorphism whenever the iterated limits and colimits exist.

A filtered category is a nonempty category in which every finite diagram has a cocone. Equivalently, any two objects map to a common object and any parallel pair becomes equal after postcomposition. A weakly filtered category requires cocones only for finite connected diagrams; equivalently, each connected component is filtered.

Write a weakly filtered category as the disjoint union ${\coprod }_s{\mathcal{J}}_s$ of its filtered connected components. Its colimit is the coproduct of the filtered colimits over the ${\mathcal{J}}_s$. In sets, a connected finite limit commutes with coproducts: connectedness forces all coordinates of a compatible tuple to lie in the same coproduct summand. By the assumed theorem, the filtered colimit over each ${\mathcal{J}}_s$ commutes with every finite limit. Applying these two facts successively proves that weakly filtered colimits commute with connected finite limits in $\mathbf{Set}$.

The forgetful functor from abelian group to sets creates finite limits and filtered colimits and reflects isomorphisms. The comparison map for a filtered colimit and a finite limit therefore becomes the corresponding isomorphism of sets, so filtered colimits commute with finite limits in $\mathbf{AbGp}$.

The dual claim fails because inverse limit need not preserve epimorphisms. Take the inverse systemswith identity bonding maps on $A_n$, reduction maps on $B_n$, and levelwise epimorphisms $q_n:A_n\to B_n$. Thenand the induced map $\mathbb{Z}\to {\mathbb{Z}}_2$ is not surjective. Since an epimorphism in abelian groups is a finite-colimit cokernel, cofiltered limits do not commute with finite colimits in $\mathbf{AbGp}$.

A category is weakly filtered when every finite connected diagram in it admits a cocone. Equivalently, each of its connected components is a filtered category.