nLab filtered colimit

Idea

Most of the earliest instances of limits and colimits used in mathematics were for diagrams indexed by the partially ordered set of natural numbers, which we now call sequential (co)limits. Many of the nice features of these (co)limits apply more generally to codirected limits and directed colimits, where the indexing category is a (co)-directed set (much as sequences in topology generalise fruitfully to nets). A filtered (co)limit is a further generalisation of this, essentially removing the requirement that the indexing category be a poset while preserving the directedness aspect in a categorified way.

Another very important class of early limits and colimits involved situations that generalised intersections and unions. If one is looking at a family of subsets of some set, then one can close it up under finite intersections and/or unions (if they are not already included) and use it to index diagrams. For instance, the family of continuous functions defined on open neighbourhoods of some point in a topological space will have this property. It was noticed that these limits and colimits behaved very nicely and a closer look showed that it was the (co)filtering nature of the indexing category that was the key. This also leads us to filtered (co)limits.

So, a filtered colimit is a colimit over a diagram from a filtered category, and a cofiltered limit (sometimes called a filtered limit) is a limit over a diagram from a cofiltered category. Taken in a suitable category such as Set, a colimit being filtered is equivalent to its commuting with finite limits.

More generally, for $\kappa$ a regular cardinal, a $\kappa$-filtered colimit is one over a $\kappa$-filtered category (and dually), and when taken with values in Set these are precisely the colimits that commute with $\kappa$-small limits.

Definition

Definition

A filtered colimit or finitely filtered colimit is a colimit of a functor $F\colon D \to C$ where $D$ is a filtered category.

For $\kappa$ a regular cardinal a $\kappa$-filtered colimit is one over a $\kappa$-filtered category.

Similarly, a cofiltered limit is a limit of a functor $F\colon D \to C$ where $D$ is a cofiltered category, or equivalently of a contravariant functor $F\colon D \to C$ (that is a functor $F\colon D^{op} \to C$) where $D$ is a filtered category.

Remark

A cofiltered limit may also be called a filtered limit (although this can be unclear); the respective terms filtered direct limit and filtered inverse limit are also popular.

A functor that preserves all finitely filtered colimits is called a finitary functor .

Properties

$\kappa$-Filtered colimits commute with $\kappa$-small limits

The following is the crucial property of filtered colimits: that they commute with finite limits.

Remark

For $C$ and $D$ two diagram categories and

$$F \,\colon\, C \times D \longrightarrow Set$$

a diagram, there is a canonical morphism

$$\lambda \,\colon\, \underset{\underset{C}{\longrightarrow}}{\lim} \; \underset{\underset{D}{\longleftarrow}}{\lim} \, F \longrightarrow \underset{\underset{D}{\longleftarrow}}{\lim} \; \underset{\underset{C}{\longrightarrow}}{\lim} \, F$$

from the colimit over $C$ of the limit over $D$ to the limit over $D$ of the colimit over $C$ of $F$:

$\lambda$ is given by a cone, whose components

$$\lambda_d \,\colon\, \underset{\underset{C}{\longrightarrow}}{\lim} \; \underset{\underset{D}{\longleftarrow}}{\lim} \, F \longrightarrow \underset{\underset{C}{\longrightarrow}}{\lim} \, F(-,d)$$

are in turn given by a cocone with components

$$(\lambda_d)_c \,\colon\, \underset{\underset{D}{\longleftarrow}}{\lim} \, F(c,-) \longrightarrow \underset{\underset{C}{\longrightarrow}}{\lim} \, F(-,d) \,.$$

This finally take to have as components

$$\underset{\underset{D}{\longleftarrow}}{\lim} \, F(c,d) \longrightarrow F(c,d) \longrightarrow \underset{\underset{C}{\longrightarrow}}{\lim} \, F(c,d) \,.$$

One checks that this indeed implies that all the components are natural and gives the existence of the original morphism.

Notice that in general $\lambda$ is not an isomorphism.

Definition

We say the limit ${\lim_\leftarrow}_D F(-,-)$ commutes with the colimit ${\lim_\to}_C F(-,-)$ if the morphism $\lambda$ above is an isomorphism

$${\lim_\to}_C {\lim_\leftarrow}_D F \stackrel{\simeq}{\longrightarrow} {\lim_\leftarrow}_D {\lim_\to}_C F \,.$$

Proposition

In Set, filtered colimits commute with finite limits.

In fact, filtered categories $C$ are precisely those shapes of diagram categories such that colimits over them commute with all finite limits in Sets.

More generally, for $\kappa$ a regular cardinal, then $\kappa$-filtered colimits commute with $\kappa$-small limits.

Proof of the first part is in MacLane (1971), §IX.2 Thm. 1 (p. 211), Borceux (1994), theorem I2.13.4. For more discussion see also BJTS 14, and see also limits and colimits by example.

Remark

It is not true that filtered colimits and finite limits commute in any category $C$ which has them. A simple example is where $C$ is the poset of closed subspaces of the one-point compactification $\mathbb{N} \cup \{\infty\}$ of the discrete space of natural numbers. If $A = \{\infty\}$ and $B_i$ ranges over finite subsets of $\mathbb{N}$, then $A \times colim_i B_i = \{\infty\} \cap (\mathbb{N} \cup \{\infty\}) = \{\infty\}$, but $colim_i A \times B_i = colim_i \{\infty\} \cap B_i = colim_i \varnothing = \varnothing$.

According to 1.5 and 1.21 in LPAC, a category has $\kappa$-directed colimits precisely if it has $\kappa$-filtered ones, and a functor preserves $\kappa$-directed colimits iff it preserves $\kappa$-filtered ones. A proof of this result, following Adamek & Rosicky, may be found here?.

The fact that directed colimits suffice to obtain all filtered ones may be regarded as a convenience similar to the fact that all colimits can be constructed from coproducts and coequalizers. Of course, a dual result holds for codirected limits.

Flat functors and points of presheaf toposes

Let $C$ be a small category. A functor $F: C \to Set$ is flat if it is a filtered colimit of representable functors.

Equivalently, a module $F: C \to Set$ is flat if and only if the tensor product

$$- \otimes_C F: Set^{C^{op}} \to Set$$

is left exact. One may prove as a corollary that if $C$ is finitely complete, $F$ is flat if and only if it is left exact (preserves finite limits). Since this tensor product is automatically a left adjoint, we have the following basic result:

Proposition

For $C$ a small category, the category of topos points of the presheaf topos $Set^{C^{op}}$ (i.e., geometric morphisms $Set \to Set^{C^{op}}$ and natural transformations between them) is equivalent to the category of flat modules on $C$.

Description in Set, Grp, Top and alike

Elements in filtered colimits in Set and Grp are given as classes of equivalences, so called germs. Filtered limits in Set and Top are given as families of compatible elements, so called threads.

More

(More was/is to be written here, including an application to geometric realization, relation to Diaconescu's theorem, and perhaps more.)

Related concepts

• filtered category, compact object

• sifted colimit, sifted (∞,1)-colimit

• filtered (∞,1)-colimit, filtered homotopy colimit

• Kolmogorov product

Filtered colimits are also important in the theory of locally presentable and accessible categories. See also pro-object and ind-object.

References

• Saunders MacLane, §IX.2 in: Categories for the Working Mathematician, Graduate texts in mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 2.13 in: Handbook of Categorical Algebra vol 1

• Hajnal Andréka and István Németi, Direct limits and filtered colimits are strongly equivalent in all categories, Banach Center Publications 9 (1982): 75-88.

Also

• Marie Bjerrum, Peter Johnstone, Tom Leinster, William F. Sawin, Notes on commutation of limits and colimits, Theory and Applications of Categories 30 (2015), 527-532 [arXiv:1409.7860, tac:3015]