We begin with the following definition. For any two categories \(\mathcal{A},\mathcal{B}\), a functor \(\mathcal{A}\rightarrow \mathcal{B}\) is called a constant functor if every object of \(\mathcal{A}\) is sent to a fixed object of \(\mathcal{B}\), namely \(B\), and every morphism is sent to \(\id_B\). With a slight abuse of notation, we sometimes write the above constant functor as \(B:\mathcal{A}\rightarrow \mathcal{B}\).
Definition 1 In a category \(\mathcal{A}\), consider a diagram1 \(F:\mathcal{I}\rightarrow \mathcal{A}\). For an object \(A\in \mathcal{A}\) and the constant functor \(A: \mathcal{I}\rightarrow \mathcal{A}\), a natural transformation \(\lambda:A \Rightarrow F\) is called a cone over \(F\), and \(A\) is called the apex of this cone, and each \(\lambda_i:A \rightarrow F(i)\) is called a leg. A cone in \(\mathcal{A}^\op\) is called a cocone.
Fix an arbitrary diagram \(F:\mathcal{I}\rightarrow \mathcal{A}\). Given a cone with apex \(A\) over \(F\) and a morphism \(A' \rightarrow A\), we can naturally define a cone with apex \(A'\) over \(F\) through this. Therefore, if we define
\[\Cone(A, F)=\{\text{cones over $F$ with apex $A$}\}\]
then \(\Cone(-,F)\) becomes a contravariant functor from \(\mathcal{A}\) to \(\Set\). When \(\mathcal{I}\) is the category arising from the poset \((\mathbb{Z},\leq)\), illustrating Definition 1 gives the following.
That all the triangles commute follows from \(\lambda\) being a natural transformation. Now if a morphism \(A' \rightarrow A\) is given, then \(\Cone(A',F)\) is defined as in the figure below.
Therefore, when the contravariant functor \(\Cone(-,F): \mathcal{A}\rightarrow \Set\) is representable, we define the representation of this functor to be the limit of \(F\).
Definition 2 For the functor \(\Cone(-,F)\), the representation \(\lambda:\lim F \Rightarrow F\) is called the limit of \(F\).
Similarly, given a cocone over \(F\), we can define the covariant functor \(\Cone(F,-):\mathcal{A}\rightarrow \Set\) through this, and the representation of this functor is called the colimit.
Universal Property of Limits
Meanwhile, in §Representable Functors, ⁋Proposition 8 we examined how to express a representation via a universal property. Applying this to our situation, consider the category of elements of the functor \(\Cone(-,F):\mathcal{A}\rightarrow \Set\):
The objects of \(\int\Cone(-,F)\) are pairs \((B, \mu)\), where \(B\) is an object of \(\mathcal{A}\) and \(\mu\in\Cone(B,F)\).
In \(\int\Cone(-,F)\), a morphism \((B,\mu)\rightarrow (B',\mu')\) is a morphism satisfying the condition \(\Cone(f,F)(\mu')=\mu\), namely \(f:B \rightarrow B'\).
That is, \(\int\Cone(-,F)\) can be thought of as the category of cones (which may have arbitrary apexes). Now, considering the dual version of §Representable Functors, ⁋Proposition 8, the limit of \(F\), \(\lambda:\lim F\Rightarrow F\), is a terminal object of \(\int\Cone(-,F)\). For instance, unpacking this a bit more in the case of the previous example, whenever the following commutative diagram
is given, there must exist a unique \(f:B \rightarrow\lim F\) making the following diagram commute.
Of course, there is no guarantee in general that the limit (or colimit) of a given diagram exists.
Definition 3 A diagram \(F:\mathcal{I}\rightarrow \mathcal{A}\) is called small if its indexing category \(\mathcal{I}\) is small. A category \(\mathcal{A}\) in which every small diagram always has a limit is called a complete category. A category \(\mathcal{A}\) in which every small diagram always has a colimit is called a cocomplete category. A functor preserving small limits is called a continuous functor, and a functor preserving small colimits is called a cocontinuous functor.
Example 4 \(\Set\) is a complete category. A key ingredient used in proving this fact is the natural isomorphism from §Representable Functors, ⁋Example 2:
\[A\cong\Hom_\Set(\ast, A)\qquad\text{for all $A\in\Set$}\tag{1}\]
Let an arbitrary (small) diagram \(F: \mathcal{I}\rightarrow \Set\) be given. If the limit of \(F\) exists, then
\[\Cone(A,F)\cong \Hom_\Set(A,\lim F)\qquad\text{for all $A\in\Set$}\]
and in particular \(\Cone(\ast,F)\cong\Hom_\Set(\ast,\lim F)\cong \lim F\), so the only candidate for \(\lim F\) is \(\Cone(\ast,F)\). Therefore we define \(\lim F:=\Cone(\ast,F)\), and it remains to find a natural transformation \(\lambda:\lim F\Rightarrow F\) and show that this cone is universal. Since a natural transformation \(\lambda:\lim F\Rightarrow F\) is determined by functions \(\lambda_i:\lim F\rightarrow F(i)\) satisfying the commuting conditions, for this purpose we first need to think about the elements of \(\Cone(\ast,F)\).
An element of \(\Cone(\ast,F)\), \(\mu:\ast\Rightarrow F\), is likewise a collection of legs \(\mu_i:\ast\rightarrow F(i)\) satisfying the commuting conditions. However, these \(\mu_i\) can in turn be regarded as elements of \(F(i)\) by equation (1), so if for each \(i\in\mathcal{I}\) we define
\[\Cone(\ast,F)\overset{\lambda_i}{\longrightarrow}F(i);\quad (\mu:\ast\Rightarrow F)\mapsto \mu_i\in F(i)\qquad\text{for all $i\in \mathcal{I}$}\]
then collecting these into \(\lambda=(\lambda_i)_{i\in\mathcal{I}}\) forms a natural transformation \(\lambda:\lim F\Rightarrow F\). Indeed, in \(\mathcal{I}\), for any morphism \(f:i \rightarrow j\) and any \(\mu\in\Cone(\ast,F)\), the fact that \(\mu\) is a cone means \(F(f)(\mu_i)=\mu_j\), so \(F(f)\circ\lambda_i=\lambda_j\) holds.
Now let us verify that this cone is universal. Given any cone \(\mu:A\Rightarrow F\), for each \(a\in A\), the naturality of \(\mu\) gives \(F(f)(\mu_i(a))=\mu_j(a)\) for all \(f:i \rightarrow j\), so \((\mu_i(a))_{i\in\mathcal{I}}\) is an element of \(\lim F=\Cone(\ast,F)\). Thus we can define a function \(h:A \rightarrow \lim F\) by \(h(a)=(\mu_i(a))_{i\in\mathcal{I}}\), and by the definition of \(\lambda_i\), \(\lambda_i\circ h=\mu_i\) holds for all \(i\in\mathcal{I}\). Conversely, for any function \(A \rightarrow \lim F\) satisfying these conditions, its value at \(a\) has its \(i\)-th component forced to be \(\mu_i(a)\), so \(h\) is unique.
Examples of Limit Categories
Depending on the shape of the category \(\mathcal{I}\), limits are also called by various names.
Example 5(Inverse limit) Consider in particular the case where \(\mathcal{I}=\omega^\op\). Here \(\omega\) is the category defined using the poset structure on the ordinal \(\omega\), so a diagram indexed by this category has the following form:
The limit of this diagram is called the inverse limit; intuitively, drawing \(\lim F\) gives the following picture:
and since \(\lim F\) looks smaller than all the \(F(i)\), it deserves to be called a limit.
Example 6(Product) A discrete category refers to a category whose only morphisms are identity morphisms. Then in the cones \(\lambda:A\Rightarrow F\) of this diagram, the naturality of \(\lambda\) imposes no conditions at all, so they simply become families of morphisms \((\lambda_i:A \rightarrow F(i))_{i\in \mathcal{I}}\). The limit \(\pi:\lim F\Rightarrow F\) of a diagram indexed by a discrete category in this way is called a product, the \(\pi_i\) are called projections, and in this case, instead of \(\lim F\), notation such as \(\prod F(i)\) is used.
In particular, rewriting the universal property of the limit cone \(\pi:\prod F(i)\Rightarrow F\) in the case of Example 6 (Product) gives the following.
Whenever an arbitrary object of \(\mathcal{A}\), \(A\), and morphisms \(\lambda_i:A \rightarrow F(i)\) are given, there exists a unique \(A \rightarrow \prod F(i)\) such that the following diagram
commutes.
To aid understanding, restricting this example to the case of the limit of sets defined in Example 4, this universal property is exactly the universal property of the product of sets defined in [Set Theory] §Product of Sets, ⁋Theorem 3.
Example 7(Equalizer) Consider the following category
Then a diagram indexed by this category has the form
and the condition that the following diagram
is a cone means that the two conditions \(\lambda_C=f\circ\lambda_B\), \(\lambda_C=g\circ\lambda_B\) are satisfied, and therefore \(f\circ\lambda_B=g\circ\lambda_B=\lambda_C\). If \(A\) is a cone, then \(\lambda_C\) is defined from \(\lambda_B\) in this way, so one can see that the only effective information is \(\lambda_B:A \rightarrow B\) satisfying \(f\circ\lambda_B=g\circ\lambda_B\). Then the limit of this diagram means the universal one among such cones.
By appropriately combining Example 6 (Product) and Example 7 (Equalizer), we can write the limit in \(\Set\) of a diagram \(\mathcal{I}\rightarrow \Set\) as a suitable equalizer diagram. As seen in Example 4, an element \(\lambda\in\lim F=\Cone(\ast,F)\) of the limit cone is, for every morphism of \(\mathcal{I}\), \(f\), one making the following diagram
commute, namely \((\lambda_i)_{i\in\mathcal{I}}\). That is, conversely, since \(\lim F\) is the collection of elements of \(\prod F(i)\) satisfying the above condition, let us define two functions from \(\prod_{i\in\mathcal{I}}F(i)\) to \(\prod_{f\in\Hom(\mathcal{I})}F(\operatorname{cod}f)\), \(a,b\), as functions that take \(\lambda=(\lambda_i)_{i\in\mathcal{I}}\in\prod_{i\in\mathcal{I}}F(i)\) and assign:
\(a(\lambda)\), having in its \(f\in\Hom(\mathcal{I})\)-th component \(\lambda_{\operatorname{cod}f}\in F(\operatorname{cod}f)\), as the element of \(\prod_{f\in\Hom(\mathcal{I})}F(\operatorname{cod}f)\) given by \((\lambda_{\operatorname{cod}f})_f\),
\(b(\lambda)\), having in its \(f\in\Hom(\mathcal{I})\)-th component \(F(f)(\lambda_{\operatorname{dom}f})\in F(\operatorname{cod}f)\), as the element of \(\prod_{f\in\Hom(\mathcal{I})}F(\operatorname{cod}f)\) given by \((F(f)(\lambda_{\operatorname{dom}f}))_f\).
Then the commutativity of the above diagram requires precisely that these two be equal, and thus \(\lim F\) can be obtained via the following equalizer limit diagram:
Although we used properties of the category \(\Set\) in this proof, this can be appropriately rephrased in the language of category theory, and thus one can show that if an arbitrary category has products and equalizers, it also has limits.
Example 8(Fiber product) Finally, consider the following category
Then a diagram indexed by this category has the form
and the limit of this diagram is given by, among those satisfying the condition \(g\circ b=f\circ a\), the universal \(A\overset{a}{\longleftarrow} X\overset{b}{\longrightarrow}B\). This is called the fiber product and denoted \(A\times_C B\). The following diagram representing this
is called a fiber diagram, and to indicate that this is a fiber diagram we draw a corner symbol \(\lrcorner\) at the fiber product.
Replacing the above examples with colimits yields the notions of direct limit, coproduct, coequalizer, and fiber coproduct. In particular, a colimit can be expressed as a suitable coequalizer diagram involving coproducts, and since \(\Set\) has both coproducts \(\coprod\) and coequalizers, we also see that \(\Set\) is cocomplete.
Limits and \(\Hom\)
Meanwhile, for any small diagram \(F: \mathcal{I}\rightarrow \mathcal{A}\) and any \(A\in \mathcal{A}\), write the composition
as \(\Hom_\mathcal{A}(A,F-)\). Then this is a diagram from \(\mathcal{I}\) to \(\Set\). By Example 4, since \(\Set\) is a complete category, the limit \(\lim \Hom_\mathcal{A}(A,F-)=\Cone(\ast,\Hom_\mathcal{A}(A,F-))\) of this diagram exists.
Since this consists of those elements of \(\prod_{i\in \mathcal{I}}\Hom_\mathcal{A}(A, F(i))\) satisfying the following compatibility condition for all \(f:i \rightarrow j\)
this is precisely \(\Cone(A,F)\), and therefore
\[\lim \Hom_\mathcal{A}(A, F-)\cong\Cone(A,F)\]
holds. On the other hand, if \(F\) has a limit, then
\[\Cone(A,F)\cong\Hom_\mathcal{A}(A,\lim F)\]
holds, so we obtain the following theorem.
Theorem 9 If an arbitrary small diagram \(F:\mathcal{I}\rightarrow \mathcal{A}\) has a limit, then the following natural isomorphism exists:
That is, \(\Hom_\mathcal{A}(A,-):\mathcal{A}\rightarrow\Set\) is a continuous functor. The same statement holds for \(\Hom_\mathcal{A}(-, A)\) and colimits.
Associativity
Finally, we introduce the following proposition.
Proposition 10 For any bifunctor \(F: \mathcal{I}\times \mathcal{J}\rightarrow \mathcal{A}\), suppose that for each \(i\in \mathcal{I}\) the limit \(\lim_{j\in \mathcal{J}}F(i,j)\) exists, and for each \(j\in \mathcal{J}\) the limit \(\lim_{i\in \mathcal{I}}F(i,j)\) exists. If both limits
exist in \(\mathcal{A}\), then they are isomorphic.
Here the inner limits again form diagrams over \(\mathcal{I}\) and over \(\mathcal{J}\), and the outer limits are the limits of the diagrams thus obtained. For instance, given a morphism in \(\mathcal{I}\), \(f:i \rightarrow i'\), the composites \(\lim_{j}F(i,j)\rightarrow F(i,j)\overset{F(f,\id_j)}{\longrightarrow}F(i',j)\) form a cone over \(F(i',-)\), and thus the universal property of the limit determines a unique morphism \(\lim_{j}F(i,j)\rightarrow \lim_{j}F(i',j)\).
A similar theorem holds for colimits. That is, limits and colimits are each associative. However, in general one cannot interchange the order of a limit and a colimit, and the best one can generally do is to use the universal property to obtain the following canonical morphism (which need not be an isomorphism):
[Rie] Emily Riehl. Category Theory in Context. Dover Publications, 2016.
Here a diagram simply means a functor \(F: \mathcal{I} \rightarrow \mathcal{A}\). This terminology is especially intuitive because, in particular when \(\mathcal{I}\) is viewed as a category from a directed set, the functor \(F\) can be thought of as producing, with \(\mathcal{I}\) as the index set, a directed system of objects of \(\mathcal{A}\). ↩
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