Limits

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Definition of the Category of Cones

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 $B$ of $\mathcal{B}$, and every morphism is sent to $\id_B$. With a slight abuse of notation, we sometimes write this constant functor as $B:\mathcal{A}\rightarrow \mathcal{B}$.

Definition 1 Consider a diagram¹ $F:\mathcal{I}\rightarrow \mathcal{A}$ defined in a category $\mathcal{A}$. 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, while 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 over $F$ with apex $A$ and a morphism $A' \rightarrow A$, we can naturally define a cone over $F$ with apex $A'$. Thus, defining

\[\Cone(A, F)=\{\text{cones over $F$ with apex $A$}\}\]

the assignment $\Cone(-,F)$ becomes a contravariant functor from $\mathcal{A}$ to $\Set$. When $\mathcal{I}$ is the category arising from the poset $(\mathbb{Z},\leq)$, we can illustrate Definition 1 as follows.

$Cone$

The fact that all 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.

$Functoriality_of_cone$

Therefore, if the contravariant functor $\Cone(-,F): \mathcal{A}\rightarrow \Set$ is representable, we define its representation to be the limit of $F$.

Definition 2 The representation $\lambda:\lim F \Rightarrow F$ of the functor $\Cone(-,F)$ is called the limit of $F$.

Similarly, given a cocone over $F$ we can define the covariant functor $\Cone(F,-):\mathcal{A}\rightarrow \Set$, and its representation is called the colimit.

Universal Property of Limits

Meanwhile, in §Representable Functors, ⁋Proposition 8 we saw how to express a representation using a universal property. Applying this to our situation, we consider the category of elements of the functor $\Cone(-,F):\mathcal{A}\rightarrow \Set$:

Objects of $\int\Cone(-,F)$ are pairs $(B, \mu)$, where $B$ is an object of $\mathcal{A}$ and $\mu\in\Cone(B,F)$.
A morphism $(B,\mu)\rightarrow (B',\mu')$ in $\int\Cone(-,F)$ is a morphism $f:B' \rightarrow B$ satisfying $\Cone(f,F)(\mu)=\mu'$.

In other words, $\int\Cone(-,F)$ can be thought of as the category of cones (with arbitrary apex). Now considering the dual version of §Representable Functors, ⁋Proposition 8, the limit $\lambda:\lim F\Rightarrow F$ of $F$ is a terminal object in $\int\Cone(-,F)$. For instance, spelling this out a bit more in the case of the previous example, for every commutative diagram

$universal_property-1$

there must exist a unique $f:B \rightarrow\lim F$ making the following diagram commute.

$universal_property-2$

Of course, in general there is no guarantee that the limit (or colimit) of a given diagram exists.

Definition 3 A diagram $F:\mathcal{I}\rightarrow \mathcal{A}$ is called small if the 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 in proving this 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 $\Cone(\ast,F)\cong\Hom_\Set(\ast,\lim F)\cong \lim F$. Thus it remains to find a natural transformation $\lambda:\lim F\Rightarrow F$ and show that this cone is universal. A natural transformation $\lambda:\lim F\Rightarrow F$ is determined by functions $\lambda_i:\lim F\rightarrow F(i)$ satisfying the commuting conditions, so to construct it we first need to consider elements of $\Cone(\ast,F)$.

An element $\mu:\ast\Rightarrow F$ of $\Cone(\ast,F)$ is likewise a collection of legs $\mu_i:\ast\rightarrow F(i)$ satisfying the commuting conditions. But these $\mu_i$ can in turn be regarded as elements of $F(i)$ by (1), so defining for each $i\in\mathcal{I}$

\[\Cone(\ast,F)\overset{\lambda_i}{\longrightarrow}F(i);\quad (\mu:\ast\Rightarrow F)\mapsto \mu_i\in F(i)\qquad\text{for all $i\in \mathcal{I}$}\]

and assembling them gives a natural transformation $\lambda:\lim F\Rightarrow F$ satisfying the desired condition.

Examples of Limits

Depending on the shape of the category $\mathcal{I}$, limits are given various names.

Example 5 (Inverse limit) Consider the special case $\mathcal{I}=\omega^\op$. 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:

$inverse_limit$

The limit of this diagram is called the inverse limit; intuitively, drawing $\lim F$ gives the picture

$inverse_limit-2$

and since $\lim F$ looks smaller than all the $F(i)$, it is worthy of the name limit.

Example 6 (Product) A discrete category is a category whose only morphisms are identity morphisms. Then in a cone $\lambda:A\Rightarrow F$ over such a diagram, the naturality of $\lambda$ imposes no conditions at all, so it is simply a family 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 is called the product, the $\pi_i$ are called projections, and in this case we use notation such as $\prod F(i)$ in place of $\lim F$.

In the special case of Example 6 (Product), rewriting the universal property of the limit cone $\pi:\prod F(i)\Rightarrow F$ gives the following:

For any object $A$ in $\mathcal{A}$ and morphisms $\lambda_i:A \rightarrow F(i)$, there exists a unique $A \rightarrow \prod F(i)$ making the following diagram

$universal_property_of_product$

commute.

To aid understanding, restricting this example to the case of limits of sets as defined in Example 4, this universal property is exactly the universal property of the product of sets defined in §Product of Sets, ⁋Theorem 3.

Example 7 (Equalizer) Consider the following category

$equlizer_indexing_category$

A diagram indexed by this category then has the form

$equalizer_diagram$

and the statement that the following diagram

$equalizer_cone$

is a cone means that the two conditions $\lambda_C=f\circ\lambda_B$ and $\lambda_C=g\circ\lambda_B$ hold, and thus $\lambda_B\circ f=\lambda_B\circ g=\lambda_C$. If $A$ is a cone then $\lambda_C$ is determined from $\lambda_B$ in this way, so the only valid information is $\lambda_B:A \rightarrow B$ satisfying $\lambda_B\circ f=\lambda_B\circ g$. Then the limit of this diagram is the universal one among such cones.

By appropriately combining Example 6 (Product) and Example 7 (Equalizer), the limit of a diagram $\mathcal{I}\rightarrow \Set$ in $\Set$ can be written as a suitable equalizer diagram. As we saw in Example 4, an element $\lambda\in\lim F=\Cone(\ast,F)$ of the limit cone is a family $(\lambda_i)_{i\in\mathcal{I}}$ making the following diagram commute for every morphism $f$ in $\mathcal{I}$:

$equalizer_seq$

That is, in other words, $\lim F$ is the collection of elements of $\prod F(i)$ satisfying the above condition, so define two functions $a,b$ from $\prod_{i\in\mathcal{I}}F(i)$ to $\prod_{f\in\Hom(\mathcal{I})}F(\operatorname{cod}f)$ by sending $\lambda=(\lambda_i)_{i\in\mathcal{I}}\in\prod_{i\in\mathcal{I}}F(i)$ to

$a(\lambda)$, the element $(\lambda_{\operatorname{cod}f})_f$ of $\prod_{f\in\Hom(\mathcal{I})}F(\operatorname{cod}f)$ whose component at $f\in\Hom(\mathcal{I})$ is $\lambda_{\operatorname{cod}f}\in F(\operatorname{cod}f)$;
$b(\lambda)$, the element $(F(f)(\lambda_{\operatorname{dom}f}))_f$ of $\prod_{f\in\Hom(\mathcal{I})}F(\operatorname{cod}f)$ whose component at $f\in\Hom(\mathcal{I})$ is $F(f)(\lambda_{\operatorname{dom}f})\in F(\operatorname{cod}f)$.

Then the commutativity of the above diagram is exactly the requirement that these two coincide, and therefore $\lim F$ can be obtained via the following equalizer limit diagram:

$equalizer_limit_diagram$

We used properties of the category $\Set$ in this proof, but it can be rephrased appropriately in the language of category theory, and thus one can show that if an arbitrary category has products and equalizers then it also has limits.

Example 8 (Fiber product) Finally, consider the following category

$fiber_indexing_category$

A diagram indexed by this category then has the form

$fiber_diagram$

and the limit of this diagram is given by the universal $A\overset{a}{\longleftarrow} X\overset{b}{\longrightarrow}B$ satisfying $g\circ b=f\circ a$. This is called the fiber product and denoted $A\times_C B$. The following diagram representing this

$fiber_diagram_limit$

is called a fiber diagram, and to indicate that this is a fiber diagram we draw a corner symbol $\lrcorner$ at the fiber product.

Reversing the above examples to colimits gives 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 $\Hom$

On the other hand, for any small diagram $F: \mathcal{I}\rightarrow \mathcal{A}$ and any $A\in \mathcal{A}$, write the composition

\[\Hom_\mathcal{A}(A,-)\circ F:\mathcal{I}\rightarrow \mathcal{A}\rightarrow \Set\]

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.

This consists of those elements of $\prod_{i\in \mathcal{I}}\Hom_\mathcal{A}(A, F(i))$ satisfying the following compatibility condition for all $f:i \rightarrow j$:

$compatibility$

and therefore is exactly $\Cone(A,F)$, so

\[\lim \Hom_\mathcal{A}(A, F-)\cong\Cone(A,F)\]

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 diagram $F:\mathcal{I}\rightarrow \mathcal{A}$ has a limit, then the following natural isomorphism

\[\lim\Hom_\mathcal{A}(A, F-)\cong\Hom_\mathcal{A}(A,\lim F)\]

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 $F: \mathcal{I}\times \mathcal{J}\rightarrow \mathcal{A}$, if both limits

\[\lim_{i\in \mathcal{I}}\lim_{j\in \mathcal{J}}F(i,j),\qquad \lim_{j\in \mathcal{J}}\lim_{i\in \mathcal{I}} F(i,j)\]

exist in $\mathcal{A}$, then they are isomorphic.

A similar theorem holds for colimits. That is, limits and colimits each satisfy associativity. However, in general one cannot interchange the order of a limit and a colimit, and the best one can do in general is to use the universal property to obtain the following canonical morphism (which need not be an isomorphism):

\[\colim_{i\in \mathcal{I}}\lim_{j\in \mathcal{J}} F(i,j) \rightarrow \lim_{j\in \mathcal{J}}\colim_{i\in \mathcal{I}} F(i,j).\]

References

[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 when $\mathcal{I}$ is viewed as a partially ordered set, since then the functor $F$ can be thought of as producing a directed system of objects in $\mathcal{A}$ indexed by $I$. ↩

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K

Limits

Definition of the Category of Cones

Universal Property of Limits

Examples of Limits

Limits and \(\Hom\)

Associativity

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