nLab limits and colimits by example

This entry lists and discusses examples and special types of the universal constructions called limits and colimits.

It starts with very elementary and simple examples and eventually passes to more sophisticated ones.

For examples of the other kinds of universal constructions see

• examples of adjoint functors

• examples of Kan extensions

Limits and colimits of sets

In the category Set of sets, the concepts of limits and colimits reduce to the familiar operations of

• cartesian product of sets;
• disjoint union of sets;
• subsets defined by equations
• quotient sets of equivalence relations.

Limits

Terminal object

The terminal object is the limit of the empty functor $F: \emptyset \to Set$. So a terminal object of $Set$ is a set $X$ such that there is a unique function from any set to $X$. This is given by any singleton set $\{a\}$, where the unique function $Y \to \{a\}$ from any set $Y$ is the function that sends every element in $Y$ to $a$.

Product

Given two sets $A, B$, the categorical product is the limit of the diagram (with no non-trivial maps)

$$\array{ A & B }.$$

This is given by the usual product of sets, which can be constructed as the set of Kuratowski pairs

$$A \times B = \{ \{\{a\}, \{a, b\}\}: a \in A, b \in B\}.$$

We tend to write $(a, b)$ instead of $\{\{a\}, \{a, b\}\}$.

The projection maps $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$ are given by

$$\pi_1(a, b) = a, \pi_2(a, b) = b.$$

To see this satisfies the universal property of products, given any pair of maps $f: X \to A$ and $g: X \to B$, we obtain a map $(f, g): X \to A \times B$ given by

$$(f, g)(x) = (f(x), g(x)).$$

More generally, given a (possibility infinite) collection of sets $\{A_\alpha\}_{\alpha \in I}$, the product of the discrete diagram consisting of these sets is the usual product $\prod_{\alpha \in I} A_\alpha$. This can be constructed as

$$\prod_{\alpha \in I} A_\alpha = \left\{f: I \to \bigcup_{\alpha \in I} A_\alpha \mid f(\alpha) \in A_\alpha\;\text{ for all }\;\alpha \in I\right\}.$$

Equalizer

Given a pair of functions $f, g: X \to Y$, the equalizer is the limit of the diagram

$$\array{ X & \underset{g}{\overset{f}{\rightrightarrows}} & Y }.$$

The limit is given by a map $e: A \to X$ such that given any $a: B \to X$, it factors through $e$ if and only if $f \circ a = g \circ a$. In other words, $a$ factors through $e$ if and only if $\im a \subseteq \{x \in X: f(x) = g(x)\}$. Thus the limit of the diagram is given by

$$A = \{x \in X: f(x) = g(x)\},$$

and the map $e: A \to X$ is given by the inclusion.

Pullback

Given two maps $f: A \to C$ and $g: B \to C$, the pullback is the limit of the diagram

$$\array{ & & A\\ & & \downarrow^\mathrlap{f}\\ B & \underset{g}{\to} & C }.$$

This limit is given by

$$\{(a, b) \in A \times B: f(a) = g(b)\},$$

with the maps to $A$ and $B$ given by the projections.

While the definition of a pullback is symmetric in $f$ and $g$, it is usually convenient to think of this as pulling back $f$ along $g$ (or the other way round). This has more natural interpretations in certain special cases.

If $g: B \to C$ is the inclusion of a subset (ie. is a monomorphism), then the pullback of $f$ along $g$ is given by

$$\{a \in A: f(a) \in B\}.$$

So this is given by restricting $f$ to the elements that are mapped into $B$.

Further, if $f: A \to C$ is also the inclusion fo a subset, so that $A$ and $B$ are both subobjects of $C$, then the above formula tells us that the pullback is simply the intersection of the two subsets.

Alternatively, we can view the map $f: A \to C$ as a collection of sets indexed by elements of $C$, where the set indexed by $c \in C$ is given by $A_c = f^{-1}(c)$. Under this interpretation, pulling $f$ back along $g$ gives a collection of sets indexed by elements of $B$, where the set indexed by $b \in B$ is given b $A_{g(b)}$.

General limits

Given a general Set-valued functor $F : D \to Set$, if the limit $lim F$ exists, then by definition, for any set $A$, a function $f: A \to lim F$ is equivalent to a compatible family of maps $f_d: A \to F(d)$ for each $d \in Obj(D)$.

In particular, since an element of a set $X$ bijects with maps $1 \to X$ from the singleton $1 = \{\emptyset\}$, we have

$$lim F \cong Set (1, lim F) \cong [D, Set](const_1, F),$$

where $const_1$ is the functor that constantly takes the value $1$. Thus the limit is given by the set of natural transformations from $const_1$ to $F$.

More concretely, a compatible family of maps $1 \to F(d)$ is given by an element $s_d \in F(d)$ for each $d \in Obj(d)$, satisfying the appropriate compatibility conditions. Thus, the limit can be realized as a subset of the product $\prod_{d \in Obj(d)} F(d)$ of all objects:

$$lim F = \left\{ (s_d)_d \in \prod_d F(d) | \forall (d \stackrel{f}{\to} d') : F(f)(s_{d}) = s_{d'} \right\}.$$

Colimits

Initial object

The initial object in $Set$ is a set $X$ such that there is a unique map from $X$ to any other set. This is given by the empty set $\emptyset$.

Coproduct

(…)

Coequalizer

(…)

Pushout

(…)

General colimits

The colimit over a Set-valued functor $F : D \to Set$ is a quotient set of the disjoint union $\coprod_{d \in Obj(D)} F(d)$:

$$colim F \simeq \left(\coprod_{d\in D} F(d)\right)/_\sim \,,$$

where the equivalence relation $\sim$ is that which is generated by

$$((x \in F(d)) \sim (x' \in F(d')))\quad if \quad (\exists (f : d \to d') \quad with F(f)(x) = x') \,.$$

If $D$ is a filtered category then the resulting equivalence relation can be described as follows:

$$((x \in F(d)) \sim (x' \in F(d')))\quad iff \quad (\exists d'', (f : d \to d''), (g: d' \to d'') \quad with F(f)(x) = F(g)(x')) \,.$$

(If $D$ is not filtered, then this description doesn’t yield an equivalence relation.)

Limits and colimits of topological spaces

We discuss limits and colimits in the category Top of topological spaces.

$\,$

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits	$\phantom{AAAA}$colimits
$\,$ point space$\,$	$\,$ empty space $\,$
$\,$ product topological space $\,$	$\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$	$\,$ quotient topological space $\,$
$\,$ fiber space $\,$	$\,$ space attachment $\,$
$\,$ mapping cocylinder, mapping cocone $\,$	$\,$ mapping cylinder, mapping cone, mapping telescope $\,$
	$\,$ cell complex, CW-complex $\,$

$\,$

Definition

Let $\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}$ be a class of topological spaces, and let $S \in Set$ be a bare set. Then

• For $\{S \stackrel{f_i}{\to} S_i \}_{i \in I}$ a set of functions out of $S$, the initial topology $\tau_{initial}(\{f_i\}_{i \in I})$ is the topology on $S$ with the minimum collection of open subsets such that all $f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i$ are continuous.

• For $\{S_i \stackrel{f_i}{\to} S\}_{i \in I}$ a set of functions into $S$, the final topology $\tau_{final}(\{f_i\}_{i \in I})$ is the topology on $S$ with the maximum collection of open subsets such that all $f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))$ are continuous.

Example

For $X$ a single topological space, and $\iota_S \colon S \hookrightarrow U(X)$ a subset of its underlying set, then the initial topology $\tau_{intial}(\iota_S)$, def. , is the subspace topology, making

$$\iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X$$

a topological subspace inclusion.

Example

Conversely, for $p_S \colon U(X) \longrightarrow S$ an epimorphism, then the final topology $\tau_{final}(p_S)$ on $S$ is the quotient topology.

Proposition

Let $I$ be a small category and let $X_\bullet \colon I \longrightarrow Top$ be an $I$-diagram in Top (a functor from $I$ to $Top$), with components denoted $X_i = (S_i, \tau_i)$, where $S_i \in Set$ and $\tau_i$ a topology on $S_i$. Then:

1. The limit of $X_\bullet$ exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology, def. , for the functions $p_i$ which are the limiting cone components:

   $$\array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,.$$

   Hence

   $$\underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$$

2. The colimit of $X_\bullet$ exists and is the topological space whose underlying set is the colimit in Set of the underlying diagram of sets, and whose topology is the final topology, def. for the component maps $\iota_i$ of the colimiting cocone

   $$\array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.$$

   Hence

   $$\underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right)$$

(e.g. Bourbaki 71, section I.4)

Proof

The required universal property of $\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$ is immediate: for

$$\array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_i }$$

any cone over the diagram, then by construction there is a unique function of underlying sets $S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i$ making the required diagrams commute, and so all that is required is that this unique function is always continuous. But this is precisely what the initial topology ensures.

The case of the colimit is formally dual.

Example

The limit over the empty diagram in $Top$ is the point $\ast$ with its unique topology.

Example

For $\{X_i\}_{i \in I}$ a set of topological spaces, their coproduct $\underset{i \in I}{\sqcup} X_i \in Top$ is their disjoint union.

In particular:

Example

For $S \in Set$, the $S$-indexed coproduct of the point, $\underset{s \in S}{\coprod}\ast$, is the set $S$ itself equipped with the final topology, hence is the discrete topological space on $S$.

Example

For $\{X_i\}_{i \in I}$ a set of topological spaces, their product $\underset{i \in I}{\prod} X_i \in Top$ is the Cartesian product of the underlying sets equipped with the product topology, also called the Tychonoff product.

In the case that $S$ is a finite set, such as for binary product spaces $X \times Y$, then a sub-basis for the product topology is given by the Cartesian products of the open subsets of (a basis for) each factor space.

Example

The equalizer of two continuous functions $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the equalizer of the underlying functions of sets

$$eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y$$

(hence the largets subset of $S_X$ on which both functions coincide) and equipped with the subspace topology, example .

Example

The coequalizer of two continuous functions $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the coequalizer of the underlying functions of sets

$$S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g)$$

(hence the quotient set by the equivalence relation generated by $f(x) \sim g(x)$ for all $x \in X$) and equipped with the quotient topology, example .

Example

For

$$\array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X }$$

two continuous functions out of the same domain, then the colimit under this diagram is also called the pushout, denoted

$$\array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,.$$

(Here $g_\ast f$ is also called the pushout of $f$, or the cobase change of $f$ along $g$.) If $g$ is an inclusion, one also write $X \cup_f Y$ and calls this the attaching space.

By example the pushout/attaching space is the quotient topological space

$$X \sqcup_A Y \simeq (X\sqcup Y)/\sim$$

of the disjoint union of $X$ and $Y$ subject to the equivalence relation which identifies a point in $X$ with a point in $Y$ if they have the same pre-image in $A$.

(graphics from Aguilar-Gitler-Prieto 02)

Example

As an important special case of example , let

$$i_n \colon S^{n-1}\longrightarrow D^n$$

be the canonical inclusion of the standard (n-1)-sphere as the boundary of the standard n-disk (both regarded as topological spaces with their subspace topology as subspaces of the Cartesian space $\mathbb{R}^n$).

Then the colimit in Top under the diagram, i.e. the pushout of $i_n$ along itself,

$$\left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,,$$

is the n-sphere $S^n$:

$$\array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,.$$

(graphics from Ueno-Shiga-Morita 95)

W

Limits and colimits in a preordered set

(..)

Limits and colimits in functor categories

The main point is that limits of functors are computed objectwise. See there for more

Colimit of a representable functor

The colimit of a representable functor (with values in Set) is the point, i.e. the terminal object in Set.

One can readily see this from a universal element-style argument, by direct inspection of cocones.

$$\array{ \mathcal{C}(D,C) && \overset{ (D \overset{f}{\to}C)^\ast }{\longleftarrow} && \mathcal{C}(C,C) \\ & \searrow && \swarrow \\ && \left\{ C \overset{id}{\to}C \right\} }$$

However, this style of reasoning does not easily generalize to higher category theory. The following gives a more abstract argument that is short and generalizes. We state it in (∞,1)-category theory just for definiteness of notation:

Proposition

(homotopy colimit over a representable functor is contractible)

Let $\mathcal{C}$ be a small (∞,1)-category and consider an ∞-groupoid-valued (∞,1)-functor $F \colon \mathcal{C}^{op} \to Groupoids_\infty$ which is representable, i.e. in the image of the (∞,1)-Yoneda embedding $F \simeq y C$:

$$\array{ \mathcal{C} & \overset{ \phantom{AA} y \phantom{AA} }{\longrightarrow} & Func \big( \mathcal{C}^{op}, Groupoids_\infty \big) \\ C &\mapsto& y C \coloneqq \mathcal{C}(-,C) }$$

Then the (∞,1)-colimit over this (∞,1)-functor is contractible, i.e. is the point, the terminal object in ∞Groupoids:

$$\underset{ \underset{\mathcal{C}}{\longrightarrow} }{\lim} (y C) \;\simeq\; \ast$$

Proof

The terminal $\infty$-groupoid $\ast$ is characterized by the fact that for each $S \in Groupoids_\infty$ we have $Groupoids_\infty(\ast, S) \simeq S$. Therefore it is sufficient to show that $\underset{\longrightarrow}{\lim}\big(y C\big)$ has the same property:

$$\begin{aligned} Groupoids_\infty \left( \underset{\longrightarrow}{\lim} \big( y C \big) , S \right) & \simeq\; Func( \mathcal{C}^{op} ,\, Groupoids_\infty ) \left( y C , const S \right) \\ & \simeq\; (const S)(C) \\ & \simeq\; S \end{aligned}$$

Here the first step is the (∞,1)-adjunction

$$Func \big( \mathcal{C}^{op}, Groupoids_\infty \big) \underoverset { \underset{const}{\longleftarrow} } { \overset{ \underset{\longrightarrow}{\lim} }{\longrightarrow} } {\phantom{AA}\bot\phantom{AA}} \Groupoids_\infty$$

and the second step is the (∞,1)-Yoneda lemma

Examples of limits

In the following examples, $D$ is a small category, $C$ is any category and the limit is taken over a functor $F : D^{op} \to C$.

Simple diagrams

• the limit of the empty diagram $D = \emptyset$ in $C$ is, if it exists the terminal object;

• if $D$ is a discrete category, i.e. a category with only identity morphisms, then a diagram $F : D \to C$ is just a collection $c_i$ of objects of $C$. Its limit is the product $\prod_i c_i$ of these.

• if $D = \{a \stackrel{\to}{\to} b\}$ then $lim F$ is the equalizer of the two morphisms $F(b) \to F(a)$.

• if $D$ has an terminal object $I$ (so that $I$ is an initial object in $D^{op}$), then the limit of any $F : D^{op} \to C$ is $F(I)$.

Filtered limits

• if $D$ is a poset, then the limit over $D^{op}$ is the supremum over the $F(d)$ with respect to $(F(d) \leq F(d')) \Leftrightarrow (F(d) \stackrel{F(\leq)}{\leftarrow} F(d'))$;

• the generalization of this is where the term “limit” for categorical limit (probably) originates from: for $D$ a filtered category, hence $D^{op}$ a cofiltered category, one may think of $(d \stackrel{f}{\to} d') \mapsto (F(d) \stackrel{F(f)}{\leftarrow} F(d')$ as witnessing that $F(d')$ is “larger than” $F(d)$ in some sense, and $lim F$ is then the “largest” of all these objects, the limiting object. This interpretation is perhaps more evident for filtered colimits, where the codomain category $C$ is thought of as being the opposite $C = E^{op}$. See the motivation at ind-object.

In terms of other operations

If products and equalizers exist in $C$, then limit of $F : D^{op} \to C$ can be exhibited as a subobject of the product of the $F(d)$, namely the equalizer of

$$\prod_{d \in Obj(D)} F(d) \stackrel{\langle F(f) \circ p_{F(t(f))} \rangle_{f \in Mor(D)} }{\to} \prod_{f \in Mor(D)} F(s(f))$$

and

$$\prod_{d \in Obj(D)} F(d) \stackrel{\langle p_{F(s(f))} \rangle_{f \in Mor(D)} }{\to} \prod_{f \in Mor(D)} F(s(f)) \,.$$

See the explicit formula for the limit in Set in terms of a subset of a product set.

In particular therefore, a category has all limits already if it has all products and equalizers.

Limits in presheaf categories

Consider limits of functors $F : D^{op} \to PSh(C)$ with values in the category of presheaves over a category $C$.

Proposition

(limits of presheaves are computed objectwise)

Limits of presheaves are computed objectwise:

$$lim F : c \mapsto lim F(-)(c)$$

Here on the right the limit is over the functor $F(-)(c) : D^{op} \to Set$.

Similarly for colimits

Similarly colimits of presheaves are computed objectwise.

Proposition

The Yoneda embedding $Y : C \to PSh(C)$ commutes with small limits:

Let $F : D^{op} \to C$, then we have

$$Y(lim F) \simeq lim (Y\circ F)$$

if $lim F$ exists.

Warning The Yoneda embedding does not in general preserve colimits.

Limits in under-categories

Limits in under categories are a special case of limits in comma categories. These are explained elsewhere. It may still be useful to spell out some details for the special case of under-categories. This is what the following does.

Proposition

Limits in an under category are computed as limits in the underlying category.

Precisely: let $C$ be a category, $t \in C$ an object, and $t/C$ the corresponding under category, and $p : t/C \to C$ the obvious projection.

Let $F : D \to t/C$ be any functor. Then, if it exists, the limit over $p \circ F$ in $C$ is the image under $p$ of the limit over $F$:

$$p(\lim F) \simeq \lim (p F)$$

and $\lim F$ is uniquely characterized by $\lim (p F)$.

Proof

Over a morphism $\gamma : d \to d'$ in $D$ the limiting cone over $p F$ (which exists by assumption) looks like

$$\array{ && \lim p F \\ & \swarrow && \searrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }$$

By the universal property of the limit this has a unique lift to a cone in the under category $t/C$ over $F$:

$$\array{ && t \\ & \swarrow &\downarrow & \searrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }$$

It therefore remains to show that this is indeed a limiting cone over $F$. Again, this is immediate from the universal property of the limit in $C$. For let $t \to Q$ be another cone over $F$ in $t/C$, then $Q$ is another cone over $p F$ in $C$ and we get in $C$ a universal morphism $Q \to \lim p F$

$$\array{ && t \\ & \swarrow & \downarrow & \searrow \\ && Q \\ \downarrow & \swarrow &\downarrow & \searrow & \downarrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }$$

A glance at the diagram above shows that the composite $t \to Q \to \lim p F$ constitutes a morphism of cones in $C$ into the limiting cone over $p F$. Hence it must equal our morphism $t \to \lim p F$, by the universal property of $\lim p F$, and hence the above diagram does commute as indicated.

This shows that the morphism $Q \to \lim p F$ which was the unique one giving a cone morphism on $C$ does lift to a cone morphism in $t/C$, which is then necessarily unique, too. This demonstrates the required universal property of $t \to \lim p F$ and thus identifies it with $\lim F$.

• Remark: One often says “$p$ reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if $U: A \to C$ is monadic (i.e., has a left adjoint $F$ such that the canonical comparison functor $A \to (U F)\text{-}Alg$ is an equivalence), then $U$ both reflects and preserves limits. In the present case, the projection $p: A = t/C \to C$ is monadic, is essentially the category of algebras for the monad $T(-) = t + (-)$, at least if $C$ admits binary coproducts. (Added later: the proof is even simpler: if $U: A \to C$ is the underlying functor for the category of algebras of an endofunctor on $C$ (as opposed to algebras of a monad), then $U$ reflects and preserves limits; then apply this to the endofunctor $T$ above.)

Further resources

Pedagogical vidoes that explain limits and colimits are at

• The Catsters, General Limits and Colimits