4. Limit and Colimit

Many constructions in algebraic topology are described by their universal properties. There are two important ways (dual notions) to define new objects of such types, called the limit and colimit. In this section, we give a brief discussion of them.

Let $\mathcal{I}$ be a small category (i.e. objects form a set). Let $\mathcal{C}$ be a category. Recall that we have a functor category (Definition 1.27) $\mathsf{Fun}(\mathcal{I},\mathcal{C}),$ where objects are functors from $\mathcal{I}$ to $\mathcal{C}$, and morphisms are natural transformations. We also write$${\mathcal{C}}^{\mathcal{I}}:=\mathsf{Fun}(\mathcal{I},\mathcal{C}).$$

Definition 4.1. We define the diagonal (or constant) functor $$\Delta :\mathcal{C}\to \text{Fun}(\mathcal{I},\mathcal{C}),$$which assigns $X\in \mathcal{C}$ to the functor $\Delta (X):\mathcal{I}\to \mathcal{C}$ that sends all objects in $\mathcal{I}$ to $X$ and all morphisms to $1_X$.

Diagram

Let $\mathcal{I}$ be a diagram, with vertices and arrows. We can define a category still denoted by $\mathcal{I}$:

•	$\mathrm{Obj}(\mathcal{I})=$ vertices in the diagram $\mathcal{I}$,
•	morphisms are composites of all given arrows as well as additional identity maps.

Example 4.2. The following diagramdefines a category with three objects $\bullet ,\circ ,\star$. There is only one morphism from $\bullet$ to $\circ$, one from $\circ$ to $\star$, and one from $\bullet$ to $\star$ which is the composite of the previous two. Given an object $A\in \mathcal{C}$, the constant functor $\Delta (A):\mathcal{I}\to \mathcal{C}$ can be represented by the following data

Example 4.3. The following diagramdefines a category with three objects $\bullet ,\circ ,\star$. There is only one morphism from $\bullet$ to $\circ$ and one from $\circ$ to $\star$. There are two morphisms from $\bullet$ to $\star$, one of them is the composite of the previous two morphisms, and the other one is represented by the arrow $\bullet \to \star$.

Example 4.4. The following diagramdefines a category with two objects $\bullet ,\circ$. Morphisms from $\bullet$ to $\bullet$ contains the identity $1_{\bullet }$, the composition of $\bullet \to \circ$ and $\circ \to \bullet$ and so on.

Given a diagram $\mathcal{I}$, a functor $F:\mathcal{I}\to \mathcal{C}$ is determined by assigning vertices and arrows the corresponding objects and morphisms in $\mathcal{C}$. For example, the following datadefine a functor from $\bullet \to \circ \leftarrow \star$ to $\mathcal{C}$. Such a data will be also called a $\mathcal{I}$-shaped diagram in $\mathcal{C}$.

Limit

Definition 4.5 (Limit). Let $F:\mathcal{I}\to \mathcal{C}$ be a functor. A limit for $F$ is an object $P$ in $\mathcal{C}$ together with a natural transformation$$\tau :\Delta (P)\Rightarrow F$$such that for every object $Q$ of $\mathcal{C}$ and every natural transformation $\eta :\Delta (Q)\Rightarrow F$, there exists a unique morphism $f:Q\to P$ such that $\tau \circ \Delta (f)=\eta$. In other words, the following diagram is commutative.

For example, consider the following $\mathcal{I}$-shaped diagram in $\mathcal{C}$ which represents a functor $F:\mathcal{I}\to \mathcal{C}$Then its limit is an object $P\in \mathcal{C}$ that fits into the commutative diagramMoreover for any other object $Q$ fitting into the same commutative diagram, there exists a unique $f:Q\to P$ which makes the following diagram commutative

Proposition 4.6. Let $F:\mathcal{I}\to \mathcal{C}$ and $P_1,P_2$ be two limits of $F$ with natural transformations ${\tau }_i:\Delta (P_i)\Rightarrow F$. Then there exists a unique isomorphism $P_1\to P_2$ in $\mathcal{C}$ which makes the following diagram commutative

Proof. This follows from the universal property of the limit.

$\square$

The above proposition says that if the limit of $F$ exists, then it is unique up to a canonical isomorphism.

Definition 4.7. We denote the limit of $F:\mathcal{I}\to \mathcal{C}$ by $\mathbf{lim}F$ (if exists).

The universal property of the limit gives the adjunction$${\mathrm{Hom}}_{\mathsf{Fun}(\mathcal{I},\mathcal{C})}(\Delta (X),F)={\mathrm{Hom}}_{\mathcal{C}}(X,\mathbf{lim}F).$$This immediately leads to the following theorem.

Theorem 4.8. Let $\mathcal{C}$ be a category. Then the following are equivalent

(1)	Every $F:\mathcal{I}\to \mathcal{C}$ has a limit.
(2)	The constant functor $\Delta :\mathcal{C}\to \mathsf{Fun}(\mathcal{I},\mathcal{C})$ has a right adjoint.

In this case, the right adjoint of the constant functor is the limit.

Example 4.9 (Pullback). The limit of the diagram $X\to Z\leftarrow Y$ giveswhich is called the pullback.

In the category $\mathsf{Set}$, the pull-back exists and is given by the subset of $X\times Y$$$P=\left\{(x,y)\in X\times Y|f(x)=g(y)\right\}\subset X\times Y.$$

Example 4.10 (Tower and inverse limit). We consider the following category $\mathbb{N}$:

•	Objects of $\mathbb{N}$ are positive integers.
•	Given $m,n\in \mathbb{N}$, the morphism set ${\mathrm{Hom}}_{\mathbb{N}}(m,n)$ is empty if $m>n$ and is a single point if $m\le n$.

Let ${\mathbb{N}}^{\text{op}}$ be the opposite category of $\mathbb{N}$. A functor $F:{\mathbb{N}}^{\text{op}}\to \mathcal{C}$ is represented by the tower diagramThe limit of the tower diagram is also called the inverse limit of the tower and written as $\underleftarrow{\lim }{X}_i$:

Theorem 4.11. Letbe adjoint functors. Assume the limit of $F:\mathcal{I}\to \mathcal{D}$ exists. Then the limit of $R\circ F:\mathcal{I}\to \mathcal{C}$ also exists and is given by $$\mathbf{lim}(R\circ F)=R(\mathbf{lim}F).$$In other words, right adjoint functors preserve limits.

Proof. Let $A\in \mathcal{C}$. Assume we have a natural transformation$$\tau :\Delta (A)\Rightarrow R\circ F.$$By adjunction, this is equivalent to a natural transformation $\Delta (L(A))\Rightarrow F$.

By the universal property of limit, there exists a unique map $L(A)\to \mathbf{lim}F$ factorizing $\Delta (L(A))\Rightarrow F$ so that$$\Delta (L(A))\Rightarrow \Delta (\mathbf{lim}F)\Rightarrow F.$$By adjunction again, this is equivalent to natural transformations$$\Delta (A)\Rightarrow \Delta (R(\mathbf{lim}F))\Rightarrow R\circ F.$$This implies that $R(\mathbf{lim}F)$ is the limit of $R\circ F$.

$\square$

Remark 4.12. A functor is called continuous if it preserves all limits. This theorem says if a functor has a left adjoint, then it is continuous. Under certain conditions, the reverse is also true (Adjoint Functor Theorem).

Corollary 4.13. The forgetful functor $\mathrm{Forget}:\mathsf{Top}\to \mathsf{Set}$ preserves limit.

Proof. $\mathrm{Forget}:\mathsf{Top}\to \mathsf{Set}$ has a left adjointwhere $\text{Discrete}$ associates a set $X$ with discrete topology.

$\square$

Example 4.14. Consider the following diagram in $\mathsf{Top}$We would like to understand the pull-back $P$ of the above diagram in $\mathsf{Top}$. By Example 4.9 and Corollary 4.13, we know that the underlying set for $P$ (if exists) is$$\mathrm{Forget}(P)=\left\{(x,y)\in X\times Y|f(x)=g(y)\right\}\subset X\times Y.$$It is not hard to see that if we assign $P$ the subspace topology of the topological product $X\times Y$, then $P$ is indeed the pull-back in $\mathsf{Top}$. In particular, pull-back exists in $\mathsf{Top}$.

The next proposition says that fibrations behave well under pull-back. We leave the proof to the reader as exercise.

Proposition 4.15. Let $p:Y\to Z$ be a fibration, and $f:X\to Z$ be continuous. Consider the pull-back diagramThen $q:Q\to X$ is also a fibration. In other words, the pull-back of a fibration is a fibration.

Colimit

The notion of colimit is dual to limit.

Definition 4.16 (Colimit). Let $F:\mathcal{I}\to \mathcal{C}$. A colimit for $F$ is an object $P$ in $\mathcal{C}$ together with a natural transformation$$\tau :F\Rightarrow \Delta (P)$$such that for every object $Q$ of $\mathcal{C}$ and every natural transformation $\eta :F\Rightarrow \Delta (Q)$, there exists a unique map $f:P\to Q$ such that $\Delta (f)\circ \tau =\eta$. In other words, the following diagram is commutativeThe colimit, if exists, is unique up to a unique isomorphism, and will be denoted by $\mathbf{colim}F$.

The following theorems are dual to the limit case as well and can be proved dually.

Theorem 4.17. Let $\mathcal{C}$ be a category. Then the following are equivalent

(1)	Every $F:\mathcal{I}\to \mathcal{C}$ has a colimit
(2)	The constant functor $\Delta :\mathcal{C}\to \mathsf{Fun}(\mathcal{I},\mathcal{C})$ has a left adjoint.

In this case, the left adjoint of the constant functor is the colimits.

Theorem 4.18. Letbe adjoint functors. Assume the colimit of $F:\mathcal{I}\to \mathcal{C}$ exists. Then the colimit of $L\circ F:\mathcal{I}\to \mathcal{D}$ also exists and is given by$$\mathbf{colim}(L\circ F)=L(\mathbf{colim}F).$$In other words, left adjoint functors preserve colimit.

Remark 4.19. A functor is called co-continuous if it preserves all colimits. This theorem says if a functor has a right adjoint, then it is co-continuous. Under certain conditions, the reverse is also true (Adjoint Functor Theorem).

Corollary 4.20. The forgetful functor $\mathrm{Forget}:\mathsf{Top}\to \mathsf{Set}$ preserves colimits.

Proof. $\mathrm{Forget}:\mathsf{Top}\to \mathsf{Set}$ has a right adjointwhere $\text{Triv}$ associates a set $X$ with trivial topology (only open subsets are $\varnothing$ and $X$).

$\square$

Example 4.21 (Pushout). The colimit of the diagram $X\leftarrow Y\to Z$ givesThis colimit is called the pushout, the dual notion of pullback. It has the following universal property

Here are some examples.

•	Let $j_1:X_0\to X_1$, $j_2:X_0\to X_2$ in $\mathsf{Top}$. Their pushout is the quotient of the disjoint union $X_1\coprod X_2$ by identifying $j_1(y)\sim j_2(y),y\in X_0$. It glues $X_1,X_2$ along $X_0$ using $j_1,j_2$. For instance, the following picture is an illustration.
•	Let ${\rho }_1:H\to G_1$, ${\rho }_2:H\to G_2$ be two morphisms in $\mathsf{Grp}$, then their pushout is$$(G_1\ast G_2)/N,$$where $G_1\ast G_2$ is the free product and $N$ is the normal subgroup generated by ${\rho }_1(h){\rho }_2^{-1}(h),h\in H$.

Example 4.22 (Telescope and direct limit). A functor $F:\mathbb{N}\to \mathcal{C}$ is represented by the telescope diagramThe colimit of telescope diagram is also called the direct limit of the telescope and written as $\underrightarrow{\lim }{X}_i$:

Product

Definition 4.23. Let $\mathcal{C}$ be a category, $\{A_{\alpha }{\}}_{\alpha \in I}$ be a set of objects in $\mathcal{C}$. Their product is an object $A$ in $\mathcal{C}$ together with ${\pi }_{\alpha }:A\to A_{\alpha }$ satisfying the following universal property: for any $X$ in $\mathcal{C}$ and $f_{\alpha }:X\to A_{\alpha }$, there exists a unique morphism $f:X\to A$ such that the following diagram commutes

For product of two objects, we have the following diagram

The product is a limit. In fact, let us equip the index set $I$ with the category structure such that it has only identity morphisms. Then the data $\{A_{\alpha }{\}}_{\alpha \in I}$ is the same as a functor $F:I\to \mathcal{C}$. Their product is precisely $\mathbf{lim}F$. In particular, the product is unique up to isomorphism if it exists. We denote it by$$\prod _{\alpha \in I}{A}_{\alpha }.$$A useful consequence is that the product is preserved under right adjoint functors (like forgetful functors).

Example 4.24.

•	Let $S_{\alpha }\in \mathsf{Set}$. Then $\prod _{\alpha }{S}_{\alpha }=\{(s_{\alpha })|s_{\alpha }\in S_{\alpha }\}$ is the Cartesian product.
•	Let $X_{\alpha }\in \mathsf{Top}$. Then $\prod _{\alpha }{X}_{\alpha }$ is the Cartesian product with induced product topology. Namely, we have $X\stackrel{f}{\to }\prod _{\alpha }{X}_{\alpha }$ is continuous if and only if $\{X\stackrel{f_{\alpha }}{\to }{X}_{\alpha }\}$ are continuous for any $\alpha$.
•	Let $G_{\alpha }\in \mathsf{Grp}$. Then $\prod _{\alpha }{G}_{\alpha }$ is the Cartesian product with induced group structure, i.e.$$\prod _{\alpha }{G}_{\alpha }=\{(g_{\alpha })\mid g_{\alpha }\in G_{\alpha }\}$$with $(g_{\alpha })\cdot (g_{\alpha }^{\prime })=(g_{\alpha }\cdot g_{\alpha }^{\prime })$.

Coproduct

Definition 4.25. Let $\mathcal{C}$ be a category, $\{A_{\alpha }{\}}_{\alpha \in I}$ be a set of objects in $\mathcal{C}$. Their coproduct is an object $A$ in $\mathcal{C}$ together with $i_{\alpha }:A_{\alpha }\to A$ satisfying the following universal property: for any $X$ in $\mathcal{C}$ and $f_{\alpha }:A_{\alpha }\to X$, there exists a unique morphism $f:A\to X$ such that the following diagram commutes

The coproduct is a colimit. As in the discussion of product, the data $\{A_{\alpha }{\}}_{\alpha \in I}$ defines a functor $F:I\to \mathcal{C}$. Their coproduct is precisely $\mathbf{colim}F$, which is unique up to isomorphism if it exists. We denote it by$$\coprod _{\alpha \in I}{A}_{\alpha }.$$A useful consequence is that the coproduct is preserved under left adjoint functors (like free constructions).

Example 4.26.

•	Let $S_{\alpha }\in \mathsf{Set}$. Then $\coprod _{\alpha }{S}_{\alpha }=\{(s_{\alpha })|s_{\alpha }\in S_{\alpha }\}$ is the disjoint union of sets.
•	Let $X_{\alpha }\in \mathsf{Top}$. Then $\coprod _{\alpha }{X}_{\alpha }$ is the disjoint union of topological spaces. Clearly, continuous maps $\{X_{\alpha }\stackrel{f_{\alpha }}{\to }Y\}$ uniquely extends to $\coprod _{\alpha }{X}_{\alpha }\to Y$.
•	Let $G_{\alpha }\in \mathsf{Grp}$. Then $\coprod _{\alpha }{G}_{\alpha }$ is the free product of groups. More precisely, we have$$\coprod _{\alpha }{G}_{\alpha }:=\{\text{word of finite length: }{x}_1{x}_2\cdots x_n\mid x_i\in G_{{\alpha }_i}\}/\sim ,$$where$$x_1\cdots x_i{x}_{i+1}\cdots x_n\sim x_1\cdots (x_i\cdot x_{i+1})\cdots x_n.$$if $x_i,x_{i+1}\in G_{\alpha }$ and $(x_i\cdot x_{i+1})$ is the group production in $G_{\alpha }$. The group structure in $\coprod _{\alpha }{G}_{\alpha }$ is$$(x_1\cdots x_n)\cdot (y_1\cdots y_m):=x_1\cdots x_n{y}_1\cdots y_m.$$Given group homomorphisms $G_{\alpha }\stackrel{f_{\alpha }}{\to }H$, it uniquely determines the group homomorphism$$\begin{array}{rl}f:\coprod _{\alpha }{G}_{\alpha }& \to H\\ x_1\cdots x_n& \mapsto f_{{\alpha }_1}(x_1)\cdots f_{{\alpha }_n}(x_n).\end{array}$$This is precisely the coproduct property. When there are only finitely many $G_{\alpha }$, we will write$$\coprod _{\alpha }{G}_{\alpha }=:G_1\star G_2\star \cdots \star G_n.$$

Wedge and smash product

Definition 4.27. We define the category ${\mathsf{Top}}_{\star }$ of pointed topological space where

•	an object $(X,x_0)$ is a topological space $X$ with a based point $x_0\in X$,
•	morphisms are based continuous maps that map based point to based point.

Given a space $X$, we can define a pointed space $X_{+}$ by adding an extra point$$X_{+}=X\coprod \star ,\text{with basepoint}\star .$$This defines a functor$$({)}_{+}:\mathsf{Top}\to {\mathsf{Top}}_{\star }.$$On the other hand, we have a forgetful functor by forgetting the base point$$\mathrm{Forget}:{\mathsf{Top}}_{\star }\to \mathsf{Top}.$$They form an adjoint pairThis implies that the limit in ${\mathsf{Top}}_{\star }$ will be the same as the limit in $\mathsf{Top}$. In particular, the product of pointed spaces $\{(X_i,x_i)\}$ in ${\mathsf{Top}}_{\star }$ is the topological product$$\prod _i{X}_i,\text{with base point}\{x_i\}.$$

In ${\mathsf{Top}}_{\star }$, the coproduct of two pointed spaces $X,Y$ is the wedge product $\vee$. Specifically,$$X\vee Y=X\coprod Y/\sim$$is the quotient of the disjoint union of $X$ and $Y$ by identifying the base points $x_0\in X$ and $y_0\in Y$. The identified based point is the new based point of $X\vee Y$. In general, we have$$\underset{i\in I}{\bigvee }{X}_i=\coprod _{i\in I}{X}_i/\sim ,$$where $\sim$ again identifies all based points in $X_i$’s. In other words, $\bigvee$ is the joining of spaces at a single point.

Example 4.28. The Figure-8 in Example 3.9 can be identified with $S^1\vee S^1$.

In ${\mathsf{Top}}_{\star }$, there is another operation, called smash product $\wedge$, which will have adjunction property and play an important role in homotopy theory as we will study later. Specifically,$$X\wedge Y=X\times Y/\sim$$is the quotient of the product space $X\times Y$ under the identifications $(x,y_0)\sim (x_0,y)$ for all $x\in X,y\in Y$. The identified point is the new based point of $X\wedge Y$. Note that we can write it as the quotient$$X\wedge Y=X\times Y/X\vee Y.$$

[h]

Figure 1. Smash product of circles

Example 4.29. There is a natural homeomorphism$$S^1\wedge S^n\cong S^{n+1}.$$This implies that $S^n\wedge S^m\cong S^{n+m}$. See Figure 1 for $n=1$ case. In this case, the result, i.e. $S^2$, can be also realized by cutting the green/purple circles on the torus (where we get a square) and gluing them (the boundary of the square) into one point.

Complete and cocomplete

Definition 4.30. A category $\mathcal{C}$ is called complete (cocomplete) if for any $F\in \mathsf{Fun}(\mathcal{I},\mathcal{C})$ with $\mathcal{I}$ a small category, the limit $\mathbf{lim}F$ (the colimit $\mathbf{colim}F$) exists.

Example 4.31. $\mathsf{Set},\mathsf{Grp},\mathsf{Ab},\mathsf{Vect},\mathsf{Top}$ are complete and cocomplete.

For example, in $\mathsf{Set}$, the limit of $F:I\to \mathsf{Set}$ is given by$$\mathbf{lim}F=\left\{(x_i{)}_{i\in I}\in \prod _{i\in I}F(i)|x_j=F(f)(x_i)\text{for any}i\stackrel{f}{\to }j\right\}\subset \prod _{i\in I}F(i),$$which is a subset of $\prod _{i\in I}F(i)$. The colimit is given by$$\mathbf{colim}F=\coprod _{i\in I}F(i)/\left\{x_i\sim F(f)(x_i)\text{for any}i\stackrel{f}{\to }j,x_i\in F(i)\right\},$$which is a quotient of ${\coprod }_{i\in I}F(i)$.

For another example, we consider $\mathsf{Top}$. Since the forgetful functor $\mathsf{Top}\to \mathsf{Set}$ has both a left adjoint and a right adjoint, it preserves both limits and colimits. Given $F:I\to \mathsf{Top}$, its limit $\mathbf{lim}F$ has the same underlying set as that in $\mathsf{Set}$ above, but equipped with the induced topology from product and subspace. Similarly, the colimit $\mathbf{colim}F$ is the quotient of disjoint unions of $F(i)$ with the induced quotient topology.

Initial and terminal object

Definition 4.32. An initial/universal object of a category $\mathcal{C}$ is an object $\star$ such that for every object $X$ in $\mathcal{C}$, there exists precisely one morphism $\star \to X$. Dually, a terminal/final object $\star$ satisfies that for every object $X$ there exists precisely one morphism $X\to \star$. If an object is both initial and terminal, it is called a zero object or null object.

The defining universal property implies that the initial object and the terminal object are unique up to isomorphism if they exist.

Example 4.33. The emptyset $\varnothing$ is the initial object in $\mathsf{Set}$, and the set with a single point is the terminal object in $\mathsf{Set}$. The same is true for $\mathsf{Top}$.

The limit of a functor $F:I\to \mathcal{C}$ can be viewed as a terminal object as follows. We define a category ${\mathcal{C}}_F$

•	an object of ${\mathcal{C}}_F$ is an object $A\in \mathcal{C}$ together with a natural transformation$$\Delta (A)\Rightarrow F,$$
•	a morphism in ${\mathcal{C}}_F$ is a morphism $f:A\to B$ in $\mathcal{C}$ such that the following diagram is commutative

Then $\mathbf{lim}F$ is the terminal object in ${\mathcal{C}}_F$. A dual construction says $\mathbf{colim}F$ can be viewed as an initial object.