Geometric intuition for limits

Question

I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and more laborious than they should have been. However, I felt like most of the understanding I gained from these exercises was gone within a week. I have a copy of MacLane's "Categories for the Working Mathematician," but whenever I pick it up, I can never seem to get through more than two or three pages (except in the introduction on foundations).

A couple months ago, I was trying to use the statements found in Hartshorne about glueing schemes and morphisms and realized that these statements were inadequate for my purposes. Looking more closely, I realized that Hartshorne's hypotheses are "wrong," in roughly the same way that it is "wrong" to require, in the definition of a basis for a topology that it be closed under finite intersections. (This would, for instance, exclude the set of open balls from being a basis for $\mathbb{R}^n$.) Working through it a bit more, I realized that the "right" statement was most easily expressed by saying that a certain kind of diagram in the category of schemes has a colimit. At this point, the notion of "colimit" began to seem much more manageable: a colimit is a way of gluing objects (and morphisms).

However, I cannot think of any similar intuition for the notion of "limit." Even in the case of a fibre product, a limit can be anything from an intersection to a product, and I find it intimidating to try to think of these two very different things as a special cases of the same construction. I understand how to show that they are; it just does not make intuitive sense, somehow.

For another example, I think (and correct me if I am wrong) that the sheaf condition on a presheaf can be expressed as stating that the contravariant functor takes colimits to limits. [This is not correct as stated. See Martin Brandenburg's answer below for an explanation of why not, as well as what the correct statement is.] It seems like a statement this simple should make everything clearer, but I find it much easier to understand the definition in terms of compatible local sections gluing together. I can (I think) prove that they are the same, but by the time I get to one end of the proof, I've lost track of the other end intuitively.

Thus, my question is this: Is there a nice, preferably geometric intuition for the notion of limit? If anyone can recommend a book on category theory that they think would appeal to someone like me, that would also be appreciated.

Answer 1

I pick up your remarks about sheaves. Indeed, the sheaf condition is a very good example to get a geometric idea of a limit.

Assume that $X$ is a set and $X_i$ are subsets of $X$ whose union is $X$. Then it is clear how to characterize functions on $X$: These are simply functions on the $X_i$ which agree on the overlaps $X_i \cap X_j$. This can be formulated in a fancy way: Let $J$ be the category whose objects are the indices $i$ and pairs of such indices $(i,j)$. It should be a preorder and we have the morphisms $(i,j) \to i, (i,j) \to j$. Consider the diagram $J \to Set$, which is given by $i \mapsto X_i, (i,j) \mapsto X_i \cap X_j$. What we have remarked above says exactly that $X$ is the colimit of this diagram! In a similar fashion, open coverings can be understood as colimits in the category of topological spaces, ringed spaces or schemes. It's all about gluing morphisms.

Now what about limits? I think it is important first to understand limits in the category of sets. If $F : J \to Set$ is a small diagram, then we can consider simply the set of "compatible elements in the image" of $F$, namely

$X = \{x \in \prod_j F(j) : \forall i \to j : x_j = F(i \to j)(x_i)\}$.

A short definition would be $X = Cone(*,F)$. Observe that we have projections $X \to F(j), x \mapsto x_j$ and with these $X$ is the limit of $F$. Now the Yoneda-Lemma or just the definition of a limit tells you how you can think of a limit in an arbitrary category: That $X$ is a limit of a diagram $F : J \to C$ amounts to say that elements of $X$ .. erm we don't have any elements, so let's say morphisms $Y \to X$, naturally correspond to compatible elem... erm morphisms $Y \to F(i)$. In other words, for every $Y$, $X(Y)$ is the set-theoretic limit of the diagramm $F(Y)$. I hope that this makes clear that the concept of limits in arbitrary categories is already visible in the category of sets.

Now let $X$ be a topological space and $O(X)$ the category of open subsets of $X$; it's an preorder with respect to the inclusion. Thus a presheaf is just a functor $F$ from $O(X)^{op}$ to the category of sets (or which suitable category you like). Now open coverings can be described as certain limits in $O(X)^{op}$, i.e. colimits in $O(X)$, as above. Observe that $F$ is a sheaf if and only if $F$ preserves these limits: If $U$ is covered by $U_i$, then $F(U)$ should be the limit of the $F(U_i), F(U_i \cap U_j)$ with transition maps $F(U_i) \to F(U_i \cap U_j), F(U_j) \to F(U_i \cap U_j)$, i.e. $F(U)$ consists of compatible elements of the $F(U_i)$, meaning that the elements of $F(U_i)$ and $F(U_j)$ restrict to the same element in $F(U_i \cap U_j)$. Thus we have a perfect geometric example of a limit: the set of sections on an open set is the limit of the set of sections on the open subsets of a covering.

Somehow this view takes over to the general case: Let $F : J \to Set$ be a functor. Regard it as a presheaf on $J^{op}$, and the map induced by $i \to j$ in $J^{op}$ as a restriction $F(j) \to F(i)$. Also call the elements of $F(i)$ sections on $i$. Then the limit of $F$ consists of compatible sections. Since I've been learning algebraic geometry, I almost always think of limits in this way.

Finally it is important to remember that limit is just the dual concept of colimit. And often algebra and geometry appear dually at once, for example sections and open subsets in sheaves. If $(X_i,\mathcal{O}_{X_i})$ are ringed spaces and you want to find the colimit, well you can guess that you have to do: Take the colimit of the $X_i$ and the limit of the $\mathcal{O}_{X_i}$ (pullbacked to the colimit).

"...the sheaf condition on a presheaf can be expressed as stating that the contravariant functor takes colimits to limits"

This is not correct. The reason is that the index category can be rather wild and colimits in preorders don't care about that. In detail: Let $U : J \to O(X)^{op}$ be a small diagram. Then the limit is just the union $V$ of $U_j$. Thus $F$ preserves this limit iff sections on $V$ are sections on the $U_j$ which are compatible with respect to the restriction morphisms given by $U$. If $J$ is discrete and $U$ maps everything to the same open subset $V$ of $X$, then the compatible sections are $F(V)^J$, which is bigger than $F(V)$.

"... I have a copy of MacLane's "Categories for the Working Mathematician," but whenever I pick it up, I can never seem to get through more than two or three pages (except in the introduction on foundations"

I think this book is still one of the best introductions into category theory. It can be hard to grasp all these abstract concepts and examples, but it gets easier as soon as you get input from other areas where category theoretic ideas are omnipresent. Your example about gluing morphisms illustrates this very well.

Answer 2

The way I think about limits and colimits is in terms of the most elementary examples of each: (co)products and (co)equalizers. Since any (co)limit can be built out of these, this is technically enough, and anyway I think it does give something of a feel for what the object should be. (I'm not being very picky about details here, since I'm just trying to describe my intuition. Feel free to correct me in comments.)

(Finite) coproducts of sets and spaces are disjoint unions, of groups are free products, of vector spaces are direct sums, and so on. The common thread is that a coproduct is the object you get by (as you say) gluing together the objects you start with and (if necessary) "closing it up" to get it to still be a group/vector space/whatever. This is reflected in the universal property: the coproduct of X and Y is the object with the property that maps out of it look like pairs of maps out of X and out of Y.

If f and g are set maps from A to B, then the coequalizer is the quotient of B obtained by setting f(x) equal to g(x) for each x in A. The cokernel of a map of R-modules, for example, is the coequalizer of the map with the zero map. So coequalizers are, in general, what you get by taking the target of the map and forcing the two maps given to you to be equal by making the appropriate identifications, and if you trace through the universal property you can see that this is what it entails: maps out of the coequalizer are the same as maps h out of B for which hf=hg.

The same sort of logic can be applied to products and equalizers. Products are very familiar in the categories I've mentioned, and they're usually even called "products", so I won't belabor that point. Equalizers are a little less familiar. If f and g are maps of sets from A to B, then the equalizer of f and g is just the set of elements x in A with the property that f(x) = g(x). For example, the kernel of a map of R-modules is the equalizer of the map with 0. In other words, whereas a coequalizer takes the target and forces the maps to be equal "after" they've been applied, an equalizer takes the source and forces the maps to be equal by throwing out everything on which they aren't. This is once again reflected by the universal property: maps into the equalizer are the same as maps h into A so that fh=gh.

A general limit is an equalizer of products, so you can use this to get some intuition for what it looks like: a limit of some diagram of sets can be thought of as the subset of the product of all the sets in the diagram consisting of elements which are consistent with the arrows in the diagram, that is, whenever there's an arrow f between A and B in the diagram, applying f to the A component of your element should give you the B component. (Notice that in my description of the equalizer above, I only took elements of the source. This is because specifying the element of the target is redundant, since it's forced to be both f(x) and g(x).)

A general colimit is a coequalizer of coproducts, so you take (in the case of spaces, say) a disjoint union of all the spaces involved, and then identify them along the maps in the diagram, so this picture of limits is kind of the dual: you take a product instead of a coproduct, and you use the other way to make maps equal, that is, taking subsets rather than quotients.

Answer 3

Since you're familiar with the example of the sheaf condition, I think a nice one-liner intuition is:

A limit of a diagram is an object of matching families in that diagram.

...defined just like how, in the case of (pre)sheaves, you define a matching family of sections on a cover. A product is then the case where there's no matching condition to satisfy. An intersection (let's say of subobjects $A,B \subseteq X$) is the case where the matching condition forces all three elements to be the same (as elements of $X$), so an element of the limit is a single element of $X$ that can be seen also as an element of $A$ and as $B$.

(This answer is pretty much a sub-quotient of Martin B's earlier answer, but I think it's a useful one-liner to extract.)

Answer 4

Are you comfortable with products? Products are just limits of discrete diagrams - the only arrows are the identity arrows. Any small limit will have a unique monic arrow to the product of all of the objects in the diagram you are taking a limit over (Check this!). So in set theory land this means that all limits are just special subsets of the product - that subset which makes all of the arrows you want commute with each other. Try working this out in the category of groups, and the category of topological spaces. Intersections being a special case is no big deal - if $U \subset W$ and $V \subset W$ then $U \cap V$ is isomorphic to that subset of $U \times W$ consisting only of pairs of the form $(x,x)$. Draw the appropriate diagrams to show that this is the picture which comes out of arrow chasing as well.

Answer 5

Of course your "intuition" request can be only about Set-based category where limits and colimits are based on Set analogue.

ABout Colimit you can think as a "amalgamated" union like glueing for a descent data (see also Boubaky- Topology (I vol.)).

ABout limits, is different, limits belong to the prodoct (discrete limit), and for a "geometric" representation the dimention grow...

Anyway an elements of $Lim_{i\in I}X_i$ can be view as a coherent chain (of the some shape of $I$) of elements: just one $x_i$ for any space $X_i$, these are connected togheter by links i.e. diagram morphism maps, and the coherence means that these elements are mapped togheter by these maps.

I hope this can help you.

Answer 6

There's an enlightening example of limit coming from topology. Arguably it was one of the motivating examples for the notion of categorical limit. In general topology it is known as limit over a filter of subsets.

Consider a category $Ouv_X$ of open subsets for a topological space $X$, morphisms being the obvious inclusions (I consider it as a subcategory of $Set$). Assume that $f: X \to Y$ is a (not necessarily continuous) mapping. You have a pair of functors associated with it: $f^{-1}$ (a preimage) and $f_*$. $$f^{-1}: Ouv_Y \to Ouv_X$$ $$f_*: Ouv_X \to Ouv_Y,~ f_*(U) = \mathrm{Lim}_{V\supset f(U)} V$$

A category $Ouv$ is a preorder: you have at most one morphism between any two objects. What is a limit in a preorder? It is just the infinum! So the right-hand side in the equation defining $f_*$ just means "the smallest open subset containing the image of U". Now consider a full subcategory $\tau_p \subset Ouv_X$ $$\tau_p = \{ U\subset X \vert p\in U \}$$

This is a set of open subsets containing a given point $p\in X$. If X is hausdorff, then the limit of $\tau_p$, considered as a diagram of sets, is just the point $p$. Now consider $\mathrm{Lim}\;f_* \circ \tau_p$. In general it can be anything, but if $f$ is continuous, then it is easy to show directly that $$\mathrm{Lim}\;f_* \circ \tau_p = f(p) = f(\mathrm{Lim}\; \tau_p)$$

That's a familiar identity $f(\lim x_n) = \lim f(x_n)$: a continuous function preserves limits! This limit-preservation property can also be figured out more abstractly, 'cause iff $f$ is continuous, then $f^{-1} \rightleftharpoons f_*$ is a pair of adjoint functors, and a right adjoint functor always preserves limits.

Answer 7

This answer is sort of an analogy, I am not quite sure how to make it precise. Further, It addresses that part of the question about a fiber product being anything from an intersection to a product (so this is perhaps a narrow answer). I am also not quite sure if this is "geometric". All of this said, a fiber product is a collection of "events" along with some dependency give by the maps that define the fiber product. In the case of the targets of the two defining arrows is the terminal object, their is no dependency whatsoever. Of course, I am speaking about this as if we were doing probability theory, but these ideas should work in any category.

Answer 8

consider a truncated cylinder as a succession of circles (say of radius 1): $(C_i)_{i\in [0,1]}$, then consider this as a functor on the set (discrete category) $[0,1]$ to $Set$ (category of sets), the a limit is the class of (no necessarily continuos) curves that are graphs of function $(f, g): [0, 1]\to R^2$ contained in the cylinder (i.e. with $f(t)^2+ g(t)^2\\leq 1\ for\ 1\leq t\leq 1$). If $I$ is a poset then diagram $(C_i)_{i\in I}$ of circles, are circles by diagram funtions connection these. Then a limit is like a path ( connected iff $I$ is connected) from circle to circle that "follow" these functions that have one ond only one crossing for any cirle.

About the colimit considering the (unordered) trees maked by points of some circles connected by some diagram function, then the colimit is the set of these (maximal connected) trees.

(of course choose "circles" is only for pedagogical reasons)