Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
A directed limit (or codirected limit) is a limit of a functor whose source category is a downward-directed set.
More generally, for a regular cardinal say that a -directed set is a poset in which every subset of cardinality has an upper bound. Then a limit over a functor is called -directed limit.
If the directed set is an ordinal, one speaks of a sequential limit.
The dual notion is that of directed colimit, a limit of a functor whose source is a downward-directed set.
Note that the terminology varies. Especially in algebra, a directed limit may be called a ‘projective limit’ or ‘inverse limit’; it's also possible to distinguish these so that an inverse limit may have an arbitrary (possibly undirected) poset as its source. On the other hand, both terms are often used for arbitrary limits as an alternative to the ‘co-’ method of distinction. (The corresponding dual terms are ‘inductive limit’ and ‘direct limit’, with no ‘co-’ even though these are colimits.)
Directed (co)limits were studied in algebra (as projective and inductive limits) before the general notion of limit in category theory. The elementary definition still seen there follows.
Let be a category.
A projective system in consists of a directed set (which we will write directed-upward as usual), a family of objects of , and a family of morphisms, such that:
- is the identity morphism on ;
- is the composite .
Then a projective cone of this projective system is an object and a family of projections such that
Finally, a projective limit of the projective system is a projective cone (where both and are suppressed in the notation, each in its own way) which is universal in that, given any projective cone , there exists a unique morphism such that
(where the left-hand is from the cone and the right-hand is from the limit).
Notice that a projective system in consists precisely of a directed set and a contravariant functor from (thought of as a category) to , while a projective cone or limit of such a projective system is precisely a cone or limit of the corresponding functor. So this is a special case of limit.
As with other limits, a projective limit, if any exists at all, is unique up to a given isomorphism, so we speak of the projective limit of a given projective system.
A projective limit in algebra is usually defined as a subalgebra of a cartesian product. To be precise, consists of those elements of such that:
This can be seen as a special case of the construction of an arbitrary limit out of products and equalizers.
Directed limits over the codirected set of natural numbers, the tower-diagram,
are extremely common. Classical examples occur in the theory of Postnikov towers and also in the definition of the solenoids.
A ring of formal power series (for a field) is a projective limit of the rings (for a natural number). Here, is the category of rings, is the directed set of natural numbers, , and is induced by the quotient map (which must be proved well defined on for ).
Similarly, a ring of -adic integers (for a prime number) is a projective limit of the rings .
A set of infinite sequences is a projective limit of sets of finite sequences (which, at the level of sets, includes the above examples).