# What are your favorite concrete examples of limits or colimits that you would compute during lunch?

(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not as dreadful as they might appear?" but it was too long.)

This is a pretty basic question which may be more appropriate for another website. However, the examples I am looking for should appeal to a working mathematician. That is why I am asking here.

Tomorrow I shall have lunch with a mathematician whom notions of limits and colimits make nervous in general. He feels that calculating small examples may help him overcome his fear. More precisely, what he is asking for are examples of various small diagrams (including fancy ones, for instance with loops) in familiar categories (topological spaces, abelian groups, &c.) whose limit or colimit we could calculate together over lunch, in the hope that he would get a better understanding of what are limits and colimits when they are taken over diagrams other than those giving pullbacks or pushouts (of which he already has a feeling).

Do you know of some particular instances of diagrams, in categories familiar to the working mathematician, whose calculation of the limit or colimit seems particularly illuminating? Or at least which could help a nervous mathematician overcome his fear of general limits and colimits?

I could come up with ad hoc examples, but perhaps there is better than that?

EDIT: Sorry for the tardy edit. I went to sleep after asking the question and just woke up thinking "I should have made it Community Wiki, added a big-list tag and provided more details". Thanks for the answers so far. (By the way, I myself am unsure as to what extent this question is appropriate for MO, but if the three votes to close could be shortly explained I would nevertheless appreciate it.)

The examples should appeal to a mathematician working in geometry and topology. For some reasons he really would like to make concrete computations. It seems that he has been faced with the following situation (to which I myself have never been faced; that is why I am asking here, in the hope that someone else already has been in the same situation as his): he is given a fancy (not the most usual) small diagram in Top or Ab or whatnot and wants to compute the limit or the colimit. Somehow he is afraid of this. I feel that my question is somewhat too broad and unprecise, but this is the best I have come up with given what I was asked myself.

• ${{1 + 1 = 2}}$. Jan 17, 2014 at 16:37
• Seriously: it could be worth mentioning that products and coproducts generalize multiplication and addition of natural numbers. Jan 17, 2014 at 16:38
• What's your friend's background? Jan 17, 2014 at 18:15
• Thanks to all who have answered so far. I have edited the post accordingly, giving some more details at the end. (Background: geometry and topology.) Jan 18, 2014 at 0:10
• The three votes to close are as "too broad," but this seems like a good question to me. Jan 18, 2014 at 0:51

I'm not sure what would work for this individual, but I'd be tempted to turn this around, Jeopardy! style. That is, instead of being presented with a diagram and trying to compute its limit/colimit, take some construction and devise a diagram which naturally expresses the construction as a limit or colimit.

So for example, this might be too easy, but consider the construction $X/A$ where $A$ is a subspace of a topological space $X$. Is this naturally a limit or colimit? Well, it's a colimit, but of what? Again, this may be too easy since your friend is comfortable around pushouts. For extra credit: what is the sensible meaning of $X/\emptyset$?

Or, take the graph of a function like $y = x^2$. Can this be thought of as a limit or colimit? This time it's a limit, namely the equal-izer of two functions from $\mathbb{R}^2$ to $\mathbb{R}$. (There's a more general lesson to be learned here, that limits are generally loci of suitable equations.)

How about the localization $\mathbb{Z}[1/p]$ where we invert a prime? Perhaps a little harder, do the same for the localization $\mathbb{Z}_p$. Or (would this be too familiar?) how would you express the $p$-adics as a limit?

Or, come up with the condition that a presheaf over a space is a sheaf. This might be either too familiar or too abstract, however. It might be best to take more concrete examples like the ones above. These are all off the top of my head, though, and somewhat untested by me personally.

• As an undergraduate reading "An elementary theory of the category of sets" I was really struck by how naturally the concept of a graph arises as the limit of the diagram. The formal definition of a function in ZFC is really the graph of the function. But just dualizing everything in sight gives the "cograph" of a function. Namely, if $f:A \to B$ is a function, its cograph is the partition of $A \sqcup B$ into sets of the form ${b}\cup f^{-1}[{b}]$, which gives a perfectly good alternative definition of a function in ZFC. Jan 18, 2014 at 3:15
• Somehow this alternate definition, while the same formal object, is really the "right way" to think about a function when it is respecting equivalence classes. I think this definition of function would make the 1st isomorphism theorem somewhat more obvious for instance - simply because it emphasizes that any function partitions its domains into equivalence classes indexed by the codomain. Jan 18, 2014 at 3:18
• @StevenGubkin Yes, when we draw pictures of functions between e.g. finite sets, with bags of dots and arrows indicating the assignments, we are really drawing the cographs. For many purposes this is the right or more illuminating picture to draw. Jan 18, 2014 at 3:32
• In the spirit of Todd's "equal-izer" comment, the two standard ways of specifying a manifold are, in essence, as a limit or colimit. E.g. if you describe a sphere as the solution-set of $x^2+y^2+z^2 = 1$, you're describing it as the equalizer of two maps $\mathbb{R}^3 \to \mathbb{R}$ (and $\mathbb{R}^3$ is a product, another kind of limit). On the other hand, if you describe it as a gluing-together of patches, say two copies of the disk $D$ overlapping in a cylinder $C$, then you're describing it as the coequalizer of two maps $C\to D+D$ (and $D+D$ is a coproduct, another type of colimit). Jan 18, 2014 at 16:49

My facourite example is that the integers $\mathbb Z$ may be obtained from a pushout diagram

$$\begin{matrix} \{0,1\} & \to & \{ 0\} \\ \downarrow && \downarrow \\ \mathcal I & \to & \mathbb Z \end{matrix}$$ in the category of groupoids where $\mathcal I$ is the groupoid with two objects $0,1$ and non identity arrows $\iota:0 \to 1, \iota^{-1}:1 \to 0$ This pushout also nicely models the way the circle $S^1$ is obtained from the unit interval $[0,1]$ by identifying $0$ and $1$. Thus we see one value of the notion of colimit, of which pushout is a special case, namely to compare concepts in different categories. Also, we know most things internally about the groupoid $\mathcal I$, whereas the integers are infinite.

• I was thinking about this example too, except mine was the coequalizer in $\mathbf{Cat}$ of the two object inclusions of the terminal category $\mathbf{1}$ into the arrow category $\mathbf{2}$. The coequalizer is the monoid of natural numbers made into a 1-object category. This example appears in Lawvere's Category of Categories paper, and in Mathematics Made Difficult. Jan 18, 2014 at 21:53
• @Todd: That example is a good one, and can also be phrased as a pushout in the category $Cat$ of small categories, noting that $Cat \to Sets$ is a bifibration of categories. Thanks also for the references. Jan 20, 2014 at 18:00
• @Todd: I mention that the groupoid example of the integers was in essence in my 1967 paper on van Kampen's theorem and was in the 1968 book on Topology. Jan 21, 2014 at 15:02

It seems that he has been faced with the following situation (to which I myself have never been faced; that is why I am asking here, in the hope that someone else already has been in the same situation as his): he is given a fancy (not the most usual) small diagram in Top or Ab or whatnot and wants to compute the limit or the colimit. Somehow he is afraid of this.

Then it seems that what your friend really needs is to learn methods for computing limits or colimits! Here are some facts which should help with your friend's difficulties in various familiar categories.

• In any category, to compute arbitrary small limits and colimits it suffices to compute products and equalizers resp. coproducts and coequalizers. This is a good exercise; for a solution see, for example, this blog post.
• In $\text{Set}$, the product of a family of sets is its Cartesian product in the usual sense, and the coproduct of a family of sets is its disjoint union in the usual sense.

• In $\text{Set}$, the equalizer of a pair of functions $f, g : X \to Y$ is $\{ x \in X | f(x) = g(x) \}$, and the coequalizer is the quotient of $Y$ by the equivalence relation generated by setting $f(x) \sim g(x)$ for all $x$.

• Many categories $C$ are equipped with a forgetful functor $U : C \to \text{Set}$ with a left adjoint (the "free object" functor). It follows that $U$ preserves limits, so when limits exist in $C$ they are computed as limits in $\text{Set}$ on underlying sets. For example, this is true of

• Topological spaces, graphs
• Groups, abelian groups, rings, various other algebraic categories
• Dually, if $C$ is equipped with a forgetful functor $U$ that has a right adjoint, then $U$ preserves colimits, so when colimits exist in $C$ they must be computed as colimits in $\text{Set}$ on underlying sets. For example, this is true of

• Topological spaces, graphs
• If $C$ is a reflective subcategory of a category $D$, let $U : C \to D$ denote the inclusion and let $F : D \to C$ denote its left adjoint. By point 4, limits in $C$ can be computed in $D$. But it is in addition true that colimits in $C$ can be computed by first computing the colimit in $D$ and then applying $F$. For example, this is true of

• The inclusion of abelian groups into groups (the left adjoint is abelianization)
• The inclusion of compact Hausdorff spaces into spaces (the left adjoint is Stone-Cech compactification)
• The inclusion of sheaves into presheaves (the left adjoint is sheafification)
• To an analyst, compute the colimit along $K\subseteq\mathbb R^n$ of the spaces $C^\infty_K(\mathbb R^n)$ of smooth functions on $\mathbb R^n$ with support contained in $K$. You can build othr spaces which will ring a bell in this way, like $L^1_{\mathrm{loc}}$, and friends.

• For an algebrist, construct the localization of a commutative ring as a colimit.

• For the complex analyst, construct the space of germs of analitic functions as a colimit.

• For a more general audience, convince them that directed limits and colimits are just unions and intersections of sorts. For example, show them that a ring/module is the colimit of its finitely generated subobjects, that the cantor set is a colimit, etc.

• Do the kernel and the cokernel, of course. Generalize to equalizers and coequalizers to show that you get useful notions in more general categories which serve similar purposes.

Fundamental example: The free topological group G over a convergent sequence, in the sense of Graev.

This is the finest topological group with countably many generators, so that the generators converge to the identity. The basic behavior of colimits of Hausdorff compacta plays a crucial role in understanding this construction, sketched as follows.

Appetizer: Fix a group G and show in general that the collection of topologies on G (under which G is a topological group) forms a complete lattice under inclusion, ( key lemma: the union of any collection of topological group topologies on G is a subbasis for a topological group) ( intersections of such topologies behave much more opaquely but save that for dessert).

Main course: Let G be the free group on countably many generators $x_1$, $x_2$,...

Summon the finest topological group topology $\tau$ on G such that $x_1, x_2$,... $\rightarrow$ 1.

The appetizers ensure $\tau$ exists, but is $(G,\tau)$ sequential? Is it metrizable? Is it normal?

The answers are yes, no, and yes, but this is entirely nonobvious. However, much of the mystery is absorbed by basic theory of colimits of Hausdorff compacta.

Notably, $(G,\tau)$ is the direct limit of countably many nested sequential Hausdorff compacta, and hence normal and sequential. Moreover (by an ultimately straightforward miracle), $(G,\tau)$ X $(G,\tau)$ is a sequential space. The utility of the latter fact is that to check continuity of group multiplication, it suffices to check that group multiplication is continuous over convergent sequences.

( To justify the use of `miracle' the plausible claim that the product of two sequential spaces is sequential, is generally false. (Recall a sequential space is a topological quotient of a metric space, or (nonobviously) equivalently, a space whose closed sets are precisely those which are closed under convergent sequences)).

Dessert : Despite the apparent naturality of the construction at hand, we are surrounded by categorical pathology. Notably, fixing the countable free group G and considering the (complete) lattice1 of topological group topologies on G, and the larger (complete) lattice2 of all topologies on G, inclusion of lattice1 into lattice2 is NOT a homomorphism. (The pathology remains even if we restrict the conversation to sequential topologies).

Antacid: The root pathology is created by the general failure of colimits to commute with products. However, fixing a category and making modest adjustments to standard defintions (such as canonically refining the standard product of sequential spaces to preserve the sequential property) can create useful structures and powerful mathematics.

Group actions! Treat $$G$$ as a category $$\mathcal{G}$$. Then a $$G$$-action is a functor $$\mathcal{G}\to \mathcal{C}$$. Its colimit is the orbit space and its limit is the fixed points.

• How does this work? A functor $\rho:\mathcal G\to\mathcal C$ picks an object $X$ and for each $g\in G$ a morphism $\rho(g):X\to X$, the action of $g$. I feel like the limit and colimit are both $X$ with the identity map. Jan 19, 2021 at 21:44
• @DanielPlácido Try it with an example! It's fun! Maybe look at $\mathbb{Z}$ acting on $\mathbb{R}$ by left translation. Jan 21, 2021 at 2:05

I think that producing beautiful examples strongly depends on the background of your friend. Nevertheless, the following is the best I can find.

Let $\cal C$ be small; compute the following limits and colimits

1. $\varprojlim {\cal C}(x,-)$
2. $\varprojlim {\cal C}(-,x)$
3. $\varinjlim {\cal C}(x,-)$
4. $\varinjlim {\cal C}(-,x)$

where $\mathcal C(-,x)\colon {\cal C}^\text{op}\to \bf Set$, $\mathcal C(x,-)\colon {\cal C}\to \bf Set$ are representable hom-functor, and $x$ is any fixed object.

Envelopes and refinements are often colimits and limits respectively, and they play an important role in Functional analysis (specifically, in the theory of topological algebras).

A typical example is the Arens-Michael envelope. Intuitively it is an operation (a functor) that turns each topological algebra $$A$$ into its nearest from the outside holomorphic algebra $$\operatorname{Env}A$$. It can be defined as the colimit of the system of Banach quotient algebras of $$A$$: $$\operatorname{Env}A=\lim_{\leftarrow} A/U,$$ where $$U$$ means a submultiplicative closed convex balanced neighborhood of zero in $$A$$ $$U\cdot U\subseteq U,$$ and $$A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown$$ is the completion of the quotient algebra $$A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$$ endowed with the norm topology with the unit ball $$U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$$ (the space $$A/U$$ will be a Banach algebra with respect to the "multiplication inherited from $$A$$").

As an illustration, if $$A$$ is the algebra $${\mathcal P}(M)$$ of polynomials on an affine algebraic manifold $$M$$, then its Arens-Michael envelope $$\text{Env}A$$ is exactly the algebra $${\mathcal O}(M)$$ of holomorphic functions on $$M$$: $$\operatorname{Env}{\mathcal P}(M)={\mathcal O}(M).$$
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $$az+b$$-group is turned into its "holomorphic version": $$\operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}),$$
for $$|q|=1$$, but the formula becomes different in the case of $$|q|\ne 1$$: $$\operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal P}^\star({\mathbb C})$$
(see details in my paper). Alexei Pirkovskii, who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking

• the nearest continuous topological algebra (see also here) with the formula $$\operatorname{Env}A={\mathcal C}(M)$$
for a subalgebra $$A$$ in $${\mathcal C}(M)$$ having $$M$$ as the involutive spectrum, and

• the nearest smooth topological algebra with the formula $$\operatorname{Env}A={\mathcal C}^\infty(M)$$
for a subalgebra $$A$$ in $${\mathcal C}^\infty(M)$$ having $$M$$ as the involutive spectrum and $$T_x(M)$$ as the involutive tangent space in each point $$x\in M$$.

• Could you explain some of the more exotic notation (for non-experts)?
– ABIM
Nov 29, 2019 at 14:23
• @AIM_BLB the notion of envelope (Env) is explained in the Wikipedia article en.wikipedia.org/wiki/Envelope_(category_theory) Usually, envelopes can be computed as colimits in the categories. You can look at the papers that are mentioned here and in Wikipedia. If you have some specific questions, ask them here, I'll answer. Nov 29, 2019 at 16:08
• I don't know the Arens-Michael envelope, but I do know that many constructions described using the word "envelope" are not functorial unless one makes a dramatic adjustment. For example, neither the injective envelope/hull of a module, nor the Dedekind-MacNeille completion of a poset, nor the algebraic closure of a field, are functorial in the ordinary sense. Is the Arens-Michael envelope functorial? Nov 29, 2019 at 17:32
• Todd, yes, the Arens-Michael envelope is a functor. Actually, the term "envelope" that I use is defined in such a way that it usually is a functor, see Theorems 1.20 and 1.21 in this my paper arxiv.org/abs/1110.2013 Nov 29, 2019 at 17:54
• Colimit over a discrete category (sum)
• limit over a discrete category (product)

Another example: colimit and limit in a category which is a partially ordered set (inf and sup).

Less fancy, and maybe too number-theoretic for a non-number-theorist, still: the limit of $$\mathbb Z/p^n$$'s, with natural surjections, is the $$p$$-adic integers $$\mathbb Z_p$$. Next, think of a colimit of things like $$\mathbb Z/p^n$$ that give $$\mathbb Q_p$$.

Finding the measurable envelope $\equiv$ the space of ergodic components (with respect to a given measure) of the tail equivalence relation on an infinite product space is equivalent to finding the corresponding projective limit in an appropriate category. However, this problem would usually require a quite hefty lunch.