In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.
Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
Definition Limits and colimits in a category are defined by means of diagrams in . Formally, a diagram of shape in is a functor from to :
The category is thought of as an index category, and the diagram is thought of as indexing a collection of objects and morphisms in patterned on .
One is most often interested in the case where the category is a small or even finite category. A diagram is said to be small or finite whenever is.
Limits
Let be a diagram of shape in a category . A cone to is an object of together with a family of morphisms indexed by the objects of , such that for every morphism in , we have .
A limit of the diagram is a cone to such that for every cone to there exists a unique morphism such that for all in .
One says that the cone factors through the cone with the unique factorization . The morphism is sometimes called the mediating morphism.
Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object has to be general enough to allow any cone to factor through it; on the other hand, has to be sufficiently specific, so that only one such factorization is possible for every cone.
Limits may also be characterized as terminal objects in the category of cones to F.
It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique up to a unique isomorphism. For this reason one often speaks of the limit of F.
Colimits
The dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here:
A co-cone of a diagram is an object of together with a family of morphisms
for every object of , such that for every morphism in , we have .
A colimit of a diagram is a co-cone of such that for any other co-cone of there exists a unique morphism such that for all in .
Colimits are also referred to as universal co-cones. They can be characterized as initial objects in the category of co-cones from .
As with limits, if a diagram has a colimit then this colimit is unique up to a unique isomorphism.
Variations Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in ). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph . If we let be the free category generated by , there is a universal diagram whose image contains . The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms.
Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.
Examples
Limits
The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, φ) of a diagram F : J → C. Terminal objects. If J is the empty category there is only one diagram of shape J: the empty one (similar to the empty function in set theory). A cone to the empty diagram is essentially just an object of C. The limit of F is any object that is uniquely factored through by every other object. This is just the definition of a terminal object. Products. If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by J. The limit L of F is called the product of these objects. The cone φ consists of a family of morphisms φX : L → F(X) called the projections of the product. In the category of sets, for instance, the products are given by Cartesian products and the projections are just the natural projections onto the various factors. Powers. A special case of a product is when the diagram F is a constant functor to an object X of C. The limit of this diagram is called the Jth power of X and denoted XJ. Equalizers. If J is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms. Kernels. A kernel is a special case of an equalizer where one of the morphisms is a zero morphism. Pullbacks. Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are f : X → Z and g : Y → Z. The limit L of F is called a pullback or a fiber product. It can nicely be visualized as a commutative square:
Inverse limits. Let J be a directed set (considered as a small category by adding arrows i → j if and only if i ≤ j) and let F : Jop → C be a diagram. The limit of F is called an inverse limit or projective limit. If J = 1, the category with a single object and morphism, then a diagram of shape J is essentially just an object X of C. A cone to an object X is just a morphism with codomain X. A morphism f : Y → X is a limit of the diagram X if and only if f is an isomorphism. More generally, if J is any category with an initial object i, then any diagram of shape J has a limit, namely any object isomorphic to F(i). Such an isomorphism uniquely determines a universal cone to F. Topological limits. Limits of functions are a special case of limits of filters, which are related to categorical limits as follows. Given a topological space X, denote by F the set of filters on X, x ∈ X a point, V(x) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x. The filters F are given a small and thin category structure by adding an arrow A → B if and only if A ⊆ B. The injection becomes a functor and the following equivalence holds :
x is a topological limit of A if and only if A is a categorical limit of
Colimits
Examples of colimits are given by the dual versions of the examples above: Initial objects are colimits of empty diagrams. Coproducts are colimits of diagrams indexed by discrete categories. Copowers are colimits of constant diagrams from discrete categories. Coequalizers are colimits of a parallel pair of morphisms. Cokernels are coequalizers of a morphism and a parallel zero morphism. Pushouts are colimits of a pair of morphisms with common domain. Direct limits are colimits of diagrams indexed by directed sets.
Properties
Existence of limits A given diagram F : J → C may or may not have a limit (or colimit) in C. Indeed, there may not even be a cone to F, let alone a universal cone.
A category C is said to have limits of shape J if every diagram of shape J has a limit in C. Specifically, a category C is said to have products if it has limits of shape J for every small discrete category J (it need not have large products), have equalizers if it has limits of shape (i.e. every parallel pair of morphisms has an equalizer), have pullbacks if it has limits of shape (i.e. every pair of morphisms with common codomain has a pullback). A complete category is a category that has all small limits (i.e. all limits of shape J for every small category J).
One can also make the dual definitions. A category has colimits of shape J if every diagram of shape J has a colimit in C. A cocomplete category is one that has all small colimits.
The existence theorem for limits states that if a category C has equalizers and all products indexed by the classes Ob(J) and Hom(J), then C has all limits of shape J. In this case, the limit of a diagram F : J → C can be constructed as the equalizer of the two morphisms
given (in component form) by
There is a dual existence theorem for colimits in terms of coequalizers and coproducts. Both of these theorems give sufficient and necessary conditions for the existence of all (co)limits of shape J.
Universal property Limits and colimits are important special cases of universal constructions.
Let C be a category and let J be a small index category. The functor category CJ may be thought of as the category of all diagrams of shape J in C. The diagonal functor
is the functor that maps each object N in C to the constant functor Δ(N) : J → C that maps all objects in J to N. That is, Δ(N)(X) = N for each object X in J and Δ(N)(f) = idN for each morphism f in J.
Given a diagram F: J → C (thought of as an object in CJ), a natural transformation ψ : Δ(N) → F (which is just a morphism in the category CJ) is the same thing as a cone from N to F. To see this, first note that Δ(N)(X) = N for all X implies that the components of ψ are morphisms ψX : N → F(X), which all share the domain N. Moreover, the requirement that the cone's diagrams commute is true simply because this ψ is a natural transformation. (Dually, a natural transformation ψ : F → Δ(N) is the same thing as a co-cone from F to N.)
Therefore, the definitions of limits and colimits can then be restated in the form: A limit of F is a universal morphism from Δ to F. A colimit of F is a universal morphism from F to Δ.
Adjunctions Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of shape J has a limit in C (for J small) there exists a limit functor
which assigns each diagram its limit and each natural transformation η : F → G the unique morphism lim η : lim F → lim G commuting with the corresponding universal cones. This functor is right adjoint to the diagonal functor Δ : C → CJ. This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all cones from N to F
which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ. This fails if J is not connected. For example, if J is a discrete category, the components of the unit are the diagonal morphisms δ : N → NJ.
Dually, if every diagram of shape J has a colimit in C (for J small) there exists a colimit functor
which assigns each diagram its colimit. This functor is left adjoint to the diagonal functor Δ : C → CJ, and one has a natural isomorphism
The unit of this adjunction is the universal cocone from F to colim F. If J is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of Δ.
Note that both the limit and the colimit functors are covariant functors.
As representations of functors
One can use Hom functors to relate limits and colimits in a category C to limits in Set, the category of sets. This follows, in part, from the fact the covariant Hom functor Hom(N, –) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.
If a diagram F : J → C has a limit in C, denoted by lim F, there is a canonical isomorphism
which is natural in the variable N. Here the functor Hom(N, F–) is the composition of the Hom functor Hom(N, –) with F. This isomorphism is the unique one which respects the limiting cones.