The very advantage of algebraic topology over algebraic geometry is the existence of a segment. From a segment it's easy to construct triangles, tetrahedra, and simplices in general. There are at least three reasons why you could enjoy simplicial stuff.

Firstly, if you end up doing complex geometry, comparison theorems allow you to compute more easily the cohomology of the space. I have seen you asked: what you can do with simplicial cohomology that you can't do with sheaf cohomology? Well, if you are lucky enough, you can find a (finite) CW decomposition of the (compactified) space you are studying. If you are not comfortable with these words, just think about triangulating a surface. From algebraic topology tools you then have an explicit finite dimensional complex that compute the cohomology, that in degree n is spanned by the free group of n-dimensional cells. A very easy consequence in the "triangulating a surface" example is that you can compute the genus by just knowing how many faces, segments and points you used. You will never have something like that in algebraic geometry, because you don't have cells! Another example: the cohomology of the (real or complex) projective space is really easy from this viewpoint, but the computations with Cech cohomology in algebraic geometry takes a few pages (the one I have seen on Hartshorne, at least).

Secondly, if you ever want to do some infinity category stuff, that is entering the algebraic geometry side of Lurie, you definitely want to learn infinity categories. While a category is intuitively made of points and directed segments, an infinity category contains all the superior cells that encode "higher deformations". In other words, an infinity category *is* a particular kind of space. This is the part I enjoy the most, because I believe my way of thinking about geometry has deeply changed since I have learnt this stuff. Personal taste, anyway.

At last, there is another instance of simplicial being the higher notion of an ordinary concept. The keyword is "derived" stuff. Take for example the Verdier duality: the compact support transport functor has not generally a right adjoint in the category of sheaves. That's why you have to move to the derived category of chain complexes of sheaves. We then see a chain complex as a "deformation" of the original abelian object. Maybe you know from the Cold Kan correspondence that in many cases the (positively graded) chain complexes of abelian objects correspond to simplicial such objects. Instead of having just one d, you have different $d_i$'s, which do not commute between them. I feel that such simplicial identities are the non commutative generalizations of $d^2 =0$.

This suggests that when working with a non abelian category, the right "derived" generalization of objects in a category $C$ are simplicial objects with values in $C$. That's why one talks about simplicial rings, simplicial schemes, simplicial sheaves... There is a whole field called derived algebraic geometry that makes substantial use of such generalizations. If I am not wrong, the first time I have seen the definition of a stack was in a similar context, even if I don't remember the details since now I have a more hands-on definition. You can look up Vezzosi and Calaque. Such techniques allow to address some very delicate problems (which I rarely understood, unluckily).

11more comments