nLab directed colimit

Abstract definition

A directed colimit is a colimit $\underset{\to}\lim F$ of a functor $F\colon J \to C$ whose source category $J$ is an (upward)-directed set.

More generally, for $\kappa$ a regular cardinal say that a $\kappa$-directed set $J$ is a poset in which every subset of cardinality $\lt \kappa$ has an upper bound. Then a colimit over a functor $J \to C$ is called $\kappa$-directed colimit.

If the directed set is an ordinal, one speaks of a sequential colimit.

The dual notion is that of codirected limit, a limit of a functor whose source is a downward-directed set.

Terminology

Note that the terminology varies. Especially in algebra, a directed colimit may be called an ‘inductive limit’ or ‘direct limit’; it's also possible to distinguish these so that a direct limit may have an arbitrary (possibly undirected) poset as its source. On the other hand, both terms are often used for arbitrary colimits as an alternative terminology. (The corresponding dual terms are ‘projective limit’ and ‘inverse limit’ for limits.)

Directed (co)limits were studied in algebra (as inductive and projective limits) before the general notion of limit in category theory. The elementary definition still seen there follows.

Concrete definition

Let $C$ be a category.

An inductive system in $C$ consists of a directed set $I$, a family $(A_i)_{i: I}$ of objects of $C$, and a family $(f_{ij}: A_i \to A_j)_{i \leq j: I}$ of morphisms, such that: * $f_{ii}: A_i \to A_i$ is the identity morphism on $A_i$; * $f_{ik}: A_i \to A_k$ is the composite $f_{ij} ; f_{jk}$.

Then an inductive cone of this inductive system is an object $X$ and a family of inductions $\iota_i: A_i \to X$ such that

$$\iota_i = f_{ij} ; \iota_j .$$

Finally, an inductive limit of the inductive system is an inductive cone $\underset{\to}\lim_i A_i$ (where both $f$ and $\iota$ are suppressed in the notation, each in its own way) which is universal in that, given any inductive cone $X$, there exists a unique morphism $u\colon \underset{\to}\lim_i A_i \to X$ such that

$$\iota_i = \iota_i ; u$$

(where the left-hand $\iota$ is from the cone $X$ and the right-hand $\iota$ is from the limit).

Notice that an inductive system in $C$ consists precisely of a directed set $I$ and a (covariant) functor from $I$ (thought of as a category) to $C$, while an inductive cone or limit of such an inductive system is precisely a cocone or colimit of the corresponding functor. So this is a special case of colimit.

As with other colimits, an inductive limit, if any exists at all, is unique up to a given isomorphism, so we speak of the inductive limit of a given inductive system.

Properties

According to 1.5 and 1.21 in the book by Jiří Adámek & Jiří Rosický, a category has $\kappa$-directed colimits iff it has $\kappa$-filtered ones, and a functor preserves $\kappa$-directed colimits iff it preserves $\kappa$-filtered ones.

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.

Applications

In algebra

An inductive limit in algebra is usually defined as a quotient of a disjoint union. To be precise, $\underset{\to}\lim_i A_i$ is the disjoint union $\biguplus_{i: I} A_i$ with $x: A_i$ identified with $y: A_j$ if

$$f_{ik}(x) = f_{jk}(y)$$

for some $k$. Here it is important that $C$ is a concrete category and that $I$ is a directed set (rather than merely a poset); this construction doesn't generalise very well.

In accessible category theory

The objects of an accessible category and of a presentable category are $\kappa$-directed limits over a given set of generators.

Examples

A Pruefer group $Z_{p^\infty}$ (for $p$ a prime number) is an inductive limit of the cyclic groups $Z_{p^n}$ (for $n$ a natural number). Here, $C$ is the category of groups, $I$ is the directed set of natural numbers, $A_i = Z_{p^i}$, and $f_{ij}: A_i \to A_j$ is induced by multiplication by $p$ (which must be proved well defined on $Z_{p^i}$ for $i \leq j$).

A stalk $F_x$ (for $F$ a sheaf on a topological space $S$ and $x$ an element of $S$) is an inductive limit of $F(U)$ (for $U$ an open neighbourhood of $x$).

Related concepts

• filtered colimit