Colimits glue. What do limits do?

Question

The informal but very useful way to think of colimits is as 'gluing things'. This intuitive view is very helpful to me, and I'd like one for limits aswell, but I haven't stumbled opon such a thing anywhere so...

If colimits glue, what do limits do?

Answer 1

Limits cut out solutions to equations.

Edit: For example, this is literally true in the case of affine schemes. Consider the affine scheme $\mathbb{A}^n = \text{Spec } k[x_1, \dots x_n]$ over a field $k$, and let $f(x_1, \dots x_n) \in k[x_1, \dots x_n]$ be a polynomial. $f$ defines a map $\mathbb{A}^n \to \mathbb{A}^1$, and the scheme $V(f) = \text{Spec } k[x_1, \dots x_n]/f$ of solutions to $f$ is precisely the fiber of $f$ over $0 \in \mathbb{A}^1$, which is a special case of the pullback.

Answer 2

As an example of limits cutting out solutions to equations think of equalizers or pullbacks in $\mathsf{Set}$. Do it now, before reading on. This is the sort of example that you need to have at your fingertips.

In any complete category $\mathcal{C}$, the limit of a functor $F: \mathcal{I} \to \mathcal{C}$ can be computed as the equalizer of two maps $\prod_{\alpha \in \mathbf{Mor} \mathcal{I}}F(\mathrm{dom}\alpha) \overset{\to}{\underset{\to}{}} \prod_{i \in I} Fi$. In $\mathsf{Set}$, the two arrows are functions in $|\mathbf{Mor}\mathcal{I}|$ variables, and the equalizer is the set of all solutions to setting these two functions equal to each other.

Now remember that the categories you're thinking of are concrete categories, usually admitting a limit-preserving (even right adjoint) functor to $\mathsf{Set}$. So their limits are calculated in the same way, with some bells and whistles to specify the structure forgotten by passing to the underlying set.

Of course, these kinds of intuitions have their limitations -- most obviously, by passing to the opposite of the category you're considering, you switch limits and colimits.

Even in perfectly ordinary settings, sometimes the "cutting-out-by-equations" notion is not the easiest way (for me) to think about limits. For instance, consider the $p$-adic integers $\mathbb{Z}_p = \varprojlim \mathbb{Z}/p^n$ where the limit is taken in $\mathsf{Ab}$ over the obvious chain of quotient maps $\dots \to \mathbb{Z}/p^{n+1} \to \mathbb{Z}/p^n \to \dots$. The cut-out-by-equations description describes this as the subgroup of $\alpha \in \prod_n \mathbb Z/p^n$ such that for each $n$, $\alpha_n = p\alpha_{n+1}$. I tend to think of this as the coordinates $\alpha_n$ being "glued" together -- somehow the individual elements of the limit are glued together whereas in a colimit it's the whole space itself that's been glued together. This can even be seen in the case of pullbacks in $\mathsf{Set}$: an element of the pullback can be thought of as "glued together" from elements of the two upstairs sets, "glued together" by an equation between them downstairs. I'm not quite sure if this can be explained in some categorical way. Maybe (homotopy?) type theory could help to understand this.

I also tend not to think of $\mathbb Z_p$ as being "cut out" of $\prod_n \mathbb{Z}/p^n$ even though it is: I think the reason is that the latter space is so huge and $\mathbb{Z}_p$ is so small -- it is a subspace of very high codimension. I suppose this is no different from the fact that I don't usually explicitly think of a CW complex as a quotient of the coproduct of all its cells.

Answer 3

Sheaves provide a nice framework for getting comfortable with the notion of a limit. See my answer at the relevant MO discussion

MO/23268: Geometric intuition for limits

There you will also find many other interesting answers.

To add something "new" and explain a little bit the other answers: Compute the pullback of the two obvious maps $(S^2)^+ \to S^2 \leftarrow (S^2)^-$, where by $(S^2)^+$ I mean the upper hemisphere and by $(S^2)^-$ the lower hemisphere.