Do people use other limits than products, equalizers and pullbacks?

Question

This is a somewhat vague question. I've seen several introductions to category theory, and when someone presents (co)limits, the typical examples are always (co)products, (co)equalizers and pullbacks/pushouts. I know how these are important and pop up a lot in different contexts in many areas of mathematics. But the definition of (co)limits are very general, and I've never seen any other concrete diagrams being used. So I was wondering whether there are any other useful (co)limits that people use (of course, "useful" might very much depend on interpretation, but any examples would be interesting to me).

Answer 1

Certainly, filtered colimits (also called direct limits) and cofiltered limits (also called inverse limits) pop up a lot :

inverse limits come up for instance in the definition of the $p$-adics, or whenever you have something profinite (e.g. in infinite Galois theory);

and direct limits come up whenever you have "finite" things that are easier to understand and you want to decompose an object in terms of its "finite" subobjects. For instance any algebraic gadget (in the sense of a model of an algebraic theory) is a filtered colimit of its finitely generated subgadgets, which often allows one to reduce the study to finitely generated things. Or you can also express infinite coproducts as filtered colimits of the finite coproducts, and those are sometimes easier to understand.

Here are some concrete examples:

-inverse limits : $\mathbb Z/p^n$, with maps $\mathbb Z/p^{n+1}\to \mathbb Z/p^n$ being the canonical projection, the inverse limit is $\mathbb Z_p$, the $p$-adics, more generally for a commutative ring $R$ and an ideal $I$, $\varprojlim_n R/I^n$ is the so-called $I$-adic completion of $R$; for a field $K$, $Gal(\overline K/K) = \varprojlim_LGal(L/K)$, where $L$ runs in the finite Galois subextensions of $\overline K/K$;

-filtered colimits: $\mathbb Z/p^n$, with the inclusion $\mathbb Z/p^n\to \mathbb Z/p^{n+1}$ being induced by multiplication by $p$ (very different from the inverse system), the colimit of that is $\mathbb Z/p^\infty$ (the $p$-primary part of $\mathbb{Q/Z}$); in topological spaces, $\mathbb RP^\infty = \mathrm{colim}_n \mathbb RP^n$ : the infinite projective space is the colimit of the finite dimensional projective spaces; in vector spaces $K[x] = \mathrm{colim}_n K[x]_{\leq n}$ (space of polynomials of degree $\leq n$)

Or in topology, compactness plays the role of finiteness, and you may be happy when you can decompose some space in the form of a filtered colimit of compact spaces (e.g. nice CW complexes)

Another type of colimit that often comes up in homotopy theory is colimits over $\Delta^{op}$, the simplex category.

Other examples have been mentioned in the comments : for the definition of the stalk of a sheaf at a point, or for a left Kan extension, you have various weird types of diagrams that can come up - and sometimes you don't know, or don't even want to know what that diagram looks like, that's why it's interesting to have a general theory of (co)limits, so you can make do without knowing the specific shape of the diagram.