nLab
directed colimit

Context

Category theory

Limits and colimits

Directed colimits

Abstract definition

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

More generally, for κ\kappa a regular cardinal say that a κ\kappa-directed set JJ is a poset in which every subset of cardinality <κ\lt \kappa has an upper bound. Then a colimit over a functor JCJ \to C is called κ\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 CC be a category.

An inductive system in CC consists of a directed set II, a family (A i) i:I(A_i)_{i: I} of objects of CC, and a family (f ij:A iA j) ij:I(f_{ij}: A_i \to A_j)_{i \leq j: I} of morphisms, such that:

  • f ii:A iA if_{ii}: A_i \to A_i is the identity morphism on A iA_i;
  • f ik:A iA kf_{ik}: A_i \to A_k is the composite f ij;f jkf_{ij} ; f_{jk}.

Then an inductive cone of this inductive system is an object XX and a family of inductions ι i:A iX\iota_i: A_i \to X such that

ι i=f ij;ι j. \iota_i = f_{ij} ; \iota_j .

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

ι i=ι i;u \iota_i = \iota_i ; u

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

Notice that an inductive system in CC consists precisely of a directed set II and a (covariant) functor from II (thought of as a category) to CC, 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 κ\kappa-directed colimits iff it has κ\kappa-filtered ones, and a functor preserves κ\kappa-directed colimits iff it preserves κ\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, lim iA i\underset{\to}\lim_i A_i is the disjoint union i:IA i\biguplus_{i: I} A_i with x:A ix: A_i identified with y:A jy: A_j if

f ik(x i)=f ik(x j) f_{ik}(x_i) = f_{ik}(x_j)

for some kk. Here it is important that CC is a concrete category and that II 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 κ\kappa-directed limits over a given set of generators.

Examples

A Pruefer group Z p Z_{p^\infty} (for pp a prime number) is an inductive limit of the cyclic groups Z p nZ_{p^n} (for nn a natural number). Here, CC is the category of groups, II is the directed set of natural numbers, A i=Z p iA_i = Z_{p^i}, and f ij:A iA jf_{ij}: A_i \to A_j is induced by multiplication by pp (which must be proved well defined on Z p iZ_{p^i} for iji \leq j).

A stalk F xF_x (for FF a sheaf on a topological space SS and xx an element of SS) is an inductive limit of F(U)F(U) (for UU an open neighbourhood of xx).

Revised on June 24, 2013 10:34:20 by Tim Porter (95.147.236.99)