{"id":445,"date":"2026-10-26T03:19:04","date_gmt":"2026-10-26T02:19:04","guid":{"rendered":"https:\/\/blogs.ethz.ch\/sirawit\/?p=445"},"modified":"2026-10-26T03:19:04","modified_gmt":"2026-10-26T02:19:04","slug":"stacks","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/sirawit\/stacks\/","title":{"rendered":"Stacks"},"content":{"rendered":"\n

I actually wanted to write an introductory note on the basic Gromov\u2013Witten invariant<\/a> \"N_d\" counting the number of rational curves passing through \"3d-1\" points in a general position, and how the recursive formula <\/p>\n\n\n\n

\"N_d = \sum_{d_1+d_2 = d; d_1, d_2 > 0} N_{d_1}N_{d_2}\left(d_1^2 d_2^2 {3d-4 \choose 3d_1-2} - d_1^3 d_2 {3d-4 \choose 3d_1-1} \right)\"<\/p>\n\n\n\n

is obtained (the main reference is the classic Fulton\u2013Pandharipande<\/a>). But yes, again, I haven’t sufficiently understood the objects there, and my advisor also suggested me to try to work with stacks<\/a> instead of coarse moduli spaces as schemes<\/a>, so in this post I’ll try to build the notion of a stack from ground up. My main reference is Stacks for Everybody<\/a> by Fantechi, and I studied Grothendieck topology from Wikipedia<\/a>.<\/p>\n\n\n\n

Why Stacks? And what are Moduli Spaces?<\/h2>\n\n\n\n

I’m not really qualified to say much about the motivation, since I haven’t really learned much about moduli spaces, but let me give an attempt.<\/p>\n\n\n\n

A moduli functor is a functor \"\mathrm{Sch}_{\mathbb{C}}^{\mathrm{op}} \to \mathrm{Sets}\".<\/p>\n\n\n\n

For a scheme \"M \in \operatorname{Obj}(\mathrm{Sch}_{\mathbb{C}})\", we define the functor \"h^M \colon \mathrm{Sch}_{\mathbb{C}}^{\mathrm{op}} \to \mathrm{Sets}\" by sending object \"X\" to \"\operatorname{Mor}_{\mathrm{Sch}_{\mathbb{C}}}(X, M)\" and sending morphism \"X' \xrightarrow{f} X\" to function \"\operatorname{Mor}_{\mathrm{Sch}_{\mathbb{C}}}(X, M) \to \operatorname{Mor}_{\mathrm{Sch}_{\mathbb{C}}}(X', M)\" defined by \"g \mapsto g \circ f\".<\/p>\n\n\n\n

A moduli functor \"h\" is representable if there exists \"M \in \operatorname{Obj}(\mathrm{Sch}_{\mathbb{C}})\" such that \"h \cong h^M\" (natural isomorphism of functors). In such case we say \"M\" is a fine moduli space<\/em> for \"h\".<\/p>\n\n\n\n

The motivation of introducing moduli spaces is the attempt to classify geometric objects up to isomorphism<\/em>. For example, we may want to parametrize the lines in \"\mathbb{A}^{n+1}\" passing through the origin. In this case, \"\mathbb{P}^n\" is the moduli space of such lines. We may want to parametrize the curves of degree \"d\" in \"\mathbb{P}^3\", obtaining the Chow variety \"\mathbf{Chow}(d, \mathbb{P}^3)\", etc. These basic objects forming some kind of a configuration space<\/em> are the basic building blocks. But now we ask further: can the configuration space be viewed as an algebro-geometric object as well? For natural problems with direct solutions, we can grab this object (and it turns out to be a scheme) and we’re done. But for some other problems, there seems to be some obstruction to the existence of such an algebro-geometric object describing the configuration space.<\/p>\n\n\n\n

Moreover, oftentimes we’re not directly interested in working with the configuration spaces, but we’re more interested in working with a family of objects in the configuration space, parametrized by another object<\/em>. That is, perhaps we’re interested in a family \"(L_t)_{t \in X}\" of objects \"L_t \in M\", parametrized by \"t \in X\". The way we formalize this is that the collection \"(L_t)_{t \in X}\" should correspond to a morphism \"X \to M\" sending \"t\" to \"L_t\". In this sense, we want to say that “family of objects in \"M\" parametrized by \"X\"” should be \"\operatorname{Mor}(X, M)\", and this is why we defined \"h^M\" that way.<\/p>\n\n\n\n

Now, the problem is there is no such scheme \"M\" representing the configuration space. This happens particularly when there are nontrivial automorphisms of objects we’re interested in, and so we give up the ideal dream of getting a fine moduli space<\/em> and step back a bit, getting a coarse moduli space<\/em> instead. Instead of trying to grab \"h^M\" so that \"h \cong h^M\", we look for a pair \"(M, \Phi)\" with \"\Phi \colon h \Rightarrow h^M\" being a natural transformation of functors, such that:<\/p>\n\n\n\n

(a) The pair is initial among other possible pairs, that is, if \"(M', \Phi')\" is another pair, then there exists a unique natural transformation \"\Psi \colon h^M \Rightarrow h^{M'}\" such that this commutes:<\/p>\n\n\n

\"Rendered<\/p>\n\n\n\n

(b) The natural transformation \"\Phi\" induces a bijection on \"\operatorname{Spec}(\mathbb{C})\", i.e., \"h(\operatorname{Spec}(\mathbb{C})) \xrightarrow{\Phi_{\operatorname{Spec}(\mathbb{C})}} h^M(\operatorname{Spec}(\mathbb{C})) = M(\mathbb{C})\" is a bijection.<\/p>\n\n\n\n

This is how we accept losing some data and study the coarse moduli spaces<\/em> as schemes in their own rights. This works, in some sense, but since this loses the precision, my advisor suggested me to work with stacks instead, so now I’ll be introducing the notion of a stack and consider how it fits into the theory of moduli spaces.<\/p>\n\n\n\n

Grothendieck Topologies<\/h2>\n\n\n\n

The Development of Topology<\/h3>\n\n\n\n

In the beginning (Bachelor Year 1), I was taught about topology in \"\mathbb{R}\" in this way (if I remember correctly):<\/p>\n\n\n\n