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
In the category Set of sets, the concepts of limits and colimits reduce to the familiar operations of
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 $a$.
Given two sets $A, B$, the categorical product is the limit of the diagram (with no non-trivial maps)
This is given by the usual product of sets, which can be constructed as the set of Kuratowski pairs
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
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
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
Given a pair of functions $f, g: X \to Y$, the equalizer is the limit of the diagram
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
and the map $e: A \to X$ is given by the inclusion.
Given two maps $f: A \to C$ and $g: B \to C$, the pullback is the limit of the diagram
This limit is given by
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
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 $A_{g(b)}$.
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
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:
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 $\emptyset$.
(…)
(…)
(…)
The colimit over a Set-valued functor $F : D \to Set$ is a quotient set of the disjoint union $\coprod_{d \in Obj(D)} F(d)$:
where the equivalence relation $\sim$ is that which is generated by
If $D$ is a filtered category then the resulting equivalence relation can be described as follows:
(If $D$ is not filtered, then this description doesn’t yield an equivalence relation.)
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 $\,$ |
$\,$
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.
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
a topological subspace inclusion.
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.
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:
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:
Hence
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
Hence
(e.g. Bourbaki 71, section I.4)
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
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.
The limit over the empty diagram in $Top$ is the point $\ast$ with its unique topology.
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:
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 $S$.
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.
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
(hence the largets subset of $S_X$ on which both functions coincide) and equipped with the subspace topology, 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
(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 .
For
two continuous functions out of the same domain, then the colimit under this diagram is also called the pushout, denoted
(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
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)
As an important special case of example , let
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,
is the n-sphere $S^n$:
(graphics from Ueno-Shiga-Morita 95)
W
(..)
The colimit over 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.
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:
(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 $F \simeq y C$:
Then the (∞,1)-colimit over this (∞,1)-functor is contractible, i.e. is the point, the terminal object in ∞Groupoids:
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:
Here the first step is the (∞,1)-adjunction
and the second step is the (∞,1)-Yoneda lemma
In the following examples, $D$ is a small category, $C$ is any category and the limit is taken over a functor $F : D^{op} \to C$.
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 $F(I)$.
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.
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
and
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.
Consider limits of functors $F : D^{op} \to PSh(C)$ with values in the category of presheaves over a category $C$.
(limits of presheaves are computed objectwise)
Limits of presheaves are computed objectwise:
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.
The Yoneda embedding $Y : C \to PSh(C)$ commutes with small limits:
Let $F : D^{op} \to C$, then we have
if $lim F$ exists.
Warning The Yoneda embedding does not in general preserve colimits.
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.
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 $F$:
and $\lim F$ is uniquely characterized by $\lim (p F)$.
Over a morphism $\gamma : d \to d'$ in $D$ the limiting cone over $p F$ (which exists by assumption) looks like
By the universal property of the limit this has a unique lift to a cone in the under category $t/C$ over $F$:
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 $Q \to \lim p F$
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 $\lim F$.
Pedagogical vidoes that explain limits and colimits are at