Notes on Moduli theory, Stacks and 2-Yoneda's Lemma
Kadri lker Berktav
Abstract This note is a survey on the basic aspects of moduli theory along with some examples. In that respect, one of the purpose of this current document is to understand how the introduction of stacks circumvents the non-representability problem of the corresponding moduli functor F by using the 2-category of stacks. To this end, we shall brie y revisit the basics of 2-category theory and present a 2-categorical version of Yoneda's lemma for the "re ned" moduli functor F. Most of the material below are standard and can be found elsewhere in the literature. For an accessible introduction to moduli theory and stacks, we refer to [1,3]. For an extensive treatment to the case of moduli of curves, see [5,6,8]. Basics of 2-category theory and further discussions can be found in [7].
Contents
1 Functor of points, representable functors, and Yoneda's Lemma1
2 Moduli theory in functorial perspective3 2.1 Moduli of vector bundles of xed rank...... 6 2.2 Moduli of elliptic curves...... 7
3 2-categories and Stacks 11 3.1 A digression on 2-categories...... 11 3.2 2-category of Stacks and 2-Yoneda's Lemma...... 14
1 Functor of points, representable functors, and Yoneda's Lemma
Main aspects of a moduli problem of interest can be encoded by a certain functor, namely a moduli functor of the form F : Cop −→ Sets (1.1) where Cop is the opposite category of the category C, and Sets denotes the category of sets. Equivalently, it is just a contravariant functor from the category C to Sets. The existence of a ne moduli space corresponds to the representability of this moduli functor. More details are to be discussed below.
De nition 1.0.1. Let C be a category. For any object U in C we de ne a functor
op hU : C −→ Sets (1.2) as follows:
1. For each object X ∈ Ob(C), X 7−→ hU (X) := MorC(X,U)
f 2. For each morphism X −→ Y ,
f f ∗ X −→ Y 7−→ MorC(Y,U) −→ MorC(X,U), g 7→ g ◦ f .
This functor hU is called "the Yoneda functor" or "the functor of points".
1 As we shall see below, this functor can be used to recover any object U of a category C by understanding morphisms into it via Yoneda's Lemma. Let F un(Cop, Sets) denote the category of functors from Cop to Sets with objects being functors F : Cop −→ Sets and morphisms being natural transformations between functors, then using the de nition of hU , one can introduce the following functor as well: h : C −→ F un(Cop, Sets) (1.3) where
1. For each object U ∈ Ob(C), U 7−→ hU = MorC(·,U)
f 2. To each morphism U −→ V , h assigns a natural transformation
hU
op C hf Sets.
hV (1.4)
Here hf is de ned as follows: (a) For each object X in C, we set
hf (X): hU (X) → hV (X), g 7−→ f ◦ g. (1.5)
η (b) Given a morphism X −→ Y in C, from the associativity property of the composition map, the diagram f◦ hU (Y ) hV (Y )
◦η ◦η
hU (X) hV (X) f◦ (1.6) commutes.
De nition 1.0.2. [4] Let C, D be two categories. 1. A functor F : C → D is called fully faithful if for any objects A, B ∈ C, the map
HomC(A, B) −→ HomD(F(A), F(B)) (1.7) is a bijection of sets.
2. A functor F : C → D is called essentially surjective if for any objects D ∈ D, there exists an object A in C such that one has an isomorphism of objects
F(A) −→∼ D. (1.8)
Lemma 1.0.1. (Yoneda's Lemma) The functor h above is fully faithful. Remark 1.0.1.
1. Yoneda's lemma implies that the functor h serves as an embedding (sometimes it is also called Yoneda's embedding), and hence hU determines U up to a unique isomorphism. Therefore, one can recover any object U in C by just knowing all possible morphisms into U. In the case of the category of -schemes, for instance, it is enough to study the restriction of this SchC C functor to the full subcategory of a ne -schemes, in order to recover the scheme . AffC C U 2. Thanks to the Yoneda's embedding, one can also realize some algebro-geometric objects (like schemes, stacks, derived "spaces", etc...) as a certain functor in addition to standard ringed- space formulation. We have the following enlightening diagram [2] encoding such a functorial interpretation:
2 Schemes op ∼ Sch = CAlgk Sets 2-categorical re nements stacks simplicial enrichments Grpds n-stacks higher categorical re nements
cdga<0 Ssets k derived stacks
One way of interpreting this diagram is as follows: In the case of schemes (stacks resp.), for instance, such a functorial description implies that points of a scheme (a stack resp.) X form a set (a groupoid resp.). These kind of interpretations, in fact, suggest the name "functor of points".
op 3. The bad news is that not all functors F : C −→ Sets are of the form hU for some U in a general set-up. In other words, h is not essentially surjective in general. This in fact leads to the following de nition: De nition 1.0.3. A functor F : Cop −→ Sets is called representable if there exists M ∈ Ob(C) such that we have a natural isomorphism F ⇔ hM. That is,
F = MorSch (·, M) for some M ∈ Ob(C). (1.9) C If this is the case, then we say that F is represented by M. In the case of moduli theory, M is then called a ne moduli space. In the next section, we shall investigate the properties of M.
2 Moduli theory in functorial perspective
A moduli problem is a problem of constructing a classifying space (or a moduli space M) for certain geometric objects (such as manifolds, algebraic varieties, vector bundles etc...) up to their intrinsic symmetries. In other words, a moduli space serves as a solution space of a given moduli problem of interest. In general, the set of isomorphism classes of objects that we would like to classify may not be able to provide a su cient information to encode geometric properties of the moduli space itself. Therefore, we expect a moduli space to behave well enough to capture the underlying geometry. Thus, this expectation leads to the following wish-list for M to be declared as a " ne" moduli space: 1. M is supposed to serve as a parameter space in a sense that there must be a one-to-one correspondence between the points of M and the set of isomorphism classes of objects to be classi ed: {points of M} ↔ {isomorphism classes of objects in C} (2.1) 2. One ensures the existence of universal classifying object, say T , through which all other objects parametrized by M can also be reconstructed. This, in fact, makes the moduli space M even more sensitive to the behavior of "families" of objects on any base object B. It is manifested by a certain representative morphism B → M. That is, for any family π : X −→ B (2.2) parametrized by some base scheme B where −1 X := {Xb ∈ Ob(C): π (b) = Xb, b ∈ B}, there exits a unique morphism f : B → M such that one has the following bered product diagram: X T
π
B M f (2.3)
where X = B ×M T . That is, the family X can be uniquely obtained by pulling back the universal object T along the morphism f.
3 For an accessible overview, see [1]. Relatively complete treatments can be found in [9,3].
Remark 2.0.1. More formally, a family over a base B is a scheme X together with a morphism π : X → B of schemes where for each (closed point) b ∈ B the ber Xb is de ned as bered product
Xb = {b} ×B X X
π
{b} B ı (2.4) where ı : {b} ,→ B is the usual inclusion map. In the language of category theory, on the other hand, a moduli problem can be formalized as a certain functor F : Cop −→ Sets, (2.5) which is called a moduli functor where Cop denotes the opposite category of the category C, and Sets is the category of sets. In other words, it is just a contravariant functor from the category to . In order to make the argument more transparent, we take to be the category C Sets C SchC of C-schemes unless otherwise stated. Note that for each C-scheme U ∈ Sch, F(U) is the set of isomorphism classes (of families) parametrized by U. For each morphism f : U → V of schemes, we have a morphism F(f): F(V ) → F(U) of sets. Example 2.0.1. Given a scheme U, one can de ne F(U) := S(U)/ ∼ where S(U) is the set of families over the base scheme U S(U) := X → U : X is a scheme over U, each ber Xu is Cg ∀u ∈ U where Cg is a smooth, projective, algebraic curve of genus g. We say that two families π : X → U and π0 : Y → U over U are equivalent if and only if there exists an isomorphism f : X −→∼ Y of schemes such that the following diagram commutes:
f X Y π π0
U (2.6)
On morphisms φ : U → V , on the other hand, we have
F(φ): F(V ) → F(U), [X → V ] 7−→ [U ×V X → U] (2.7) where U ×V X is the bered product given by pulling back the family X → V along the morphism φ : U → V :
U ×V X X
U V f (2.8)
With the above formalism in hand, the existence of a ne moduli space, therefore, corresponds to the representability of the moduli functor F in the sense that
F = MorSch (·, M) for some M ∈ Sch . (2.9) C C If this is the case, then we say that F is represented by M.
4 Remark 2.0.2. Let F : Cop −→ Sets be a moduli functor represented by an object M, then one can recast the desired properties of being a " ne" moduli space as follows:
1. Take B := spec(C) = {∗}, then from the representability we have ∼ F({∗}) = hM ({∗}) = MorC({∗},M). (2.10)
Note that the RHS is just the set of (closed) points of M, and LHS is the set of corresponding isomorphism classes.
2. When B := M, then we get an isomorphism ∼ F(M) = hM (M) = MorC(M,M), (2.11)
which allows us to de ne the universal object T to be the object corresponding to the identity morphism idM ∈ MorC(M,M). These observations yield the following corollary:
Corollary 2.0.1. If F : Cop −→ Sets is a moduli functor represented by an object M in C, then there exists an one-to-one correspondence between the set {X → B} of (equivalences classes of) families and MorC(B, M). That is,
{X → B}/ ∼ ←→ MorC(B, M). (2.12)
Furthermore, for a morphism f : B → M corresponding to the equivalence class X → B of the family X → B, we have X → B = B ×M T → B . (2.13) In many cases, however, a moduli functor is not representable in the category Sch of schemes. This is the place where the notion of a stack comes into play. In that situation, one can still make sense of the notion of a moduli space in a weaker sense. This version, namely a coarse moduli space, is still e cient enough to encode the isomorphism classes of points. That is, it has the correct points, and captures the geometry of moduli space. However, the sensitivity on the behavior of arbitrary families is no longer available. In other words, a coarse moduli space may not be able to distinguish two non-isomorphic families in many cases. Hence, the classi cation in this "family-wise" level is by no means possible. To elaborate the last statement, we rst introduce the formal de nition of a so-called coarse moduli space, and then we shall provide two important examples: (i) the moduli problem of classifying vector bundles of xed rank over an algebraic curve over a eld k [3], and (ii) the moduli of elliptic curves [1,5]. De nition 2.0.1. Let for the sake of simplicity. A coarse moduli space for a moduli C := SchC op functor F : C −→ Sets consists of a pair (M, ψ) where M is an object in C , and ψ : F → hM is a natural transformation such that 1. is a bijection of sets. ψspec(C) : F(spec(C)) → hM (spec(C)) 2. Such a pair (M, ψ) satis es the following universal property: For any scheme N and any natural transformation φ : F → hN , there exists a unique morphism f : M → N such that the following diagram commutes:
ψ F hM
φ ∃hf h N (2.14)
Remark 2.0.3. Here, hf : hM → hN is the associated natural transformation of functors 1.3. Second condition also implies that if it exists, a coarse moduli space M for a moduli functor F is unique up to a unique isomorphism.
5 Proposition 2.0.1. [9] Let (M, ψ) be a coarse moduli space for a moduli functor F : Cop −→ Sets where M is a scheme and ψ : F → hM is the corresponding natural transformation. Then (M, ψ) is a ne moduli space if and only if the following conditions hold:
1. There exists a family T → M such that ψM (T ) = idM ∈ MorC(M,M). 2. For families X → B and Y → B on a base scheme B,
[X → B] = [Y → B] ⇐⇒ ψB(X) = ψB(Y ). (2.15)
Proof. It follows directly from the de nition of a ne moduli space.
2.1 Moduli of vector bundles of xed rank We would like to investigate the moduli problem of classifying vector bundles of xed rank over a smooth, projective algebraic curve X of genus g over a eld k with char k = 0. We de ne the corresponding moduli functor F n : Schop −→ Sets (2.16) X C as follows: To each object in , n assigns the set n of isomorphism classes of families U SchC FX FX (U) of vector bundles of rank n on X parametrized by U. That is,
n is a vector bundle of rank n FX (U) = E → U ×specC X : E / ∼ Here, we say that two vector bundles and 0 0 over π : E → U ×specC X π : E → U ×specC X U ×specC X are equivalent if and only if there exists an isomorphism f : E −→∼ E0 of vector bundles such that the following diagram commutes:
f E E0 π π0
U ×spec X C (2.17)
To each morphism in , n assigns the map of vector bundles f : U → V SchC FX n n n (2.18) FX (f): FX (V ) → FX (U) which is de ned by pulling back of the vector bundle along the morphism E → V ×specC X . Note that is just the usual direct product with the projection maps f × idX U ×specC X U × X pr1 : U × X → U and pr2 : U × X → X such that
pr2 U ×specC X X
pr1
U spec C (2.19)
Now, we would like to show that n can not be representable by some scheme FX M. Claim: n is not representable in . FX SchC Proof. Assume that n is representable by a scheme . That is, we have a natural isomorphism FX M n ∼ (2.20) FX = hM . Let be a scheme and n . Then, we have a vector bundle of rank . Let U E ∈ FX (U) E → U × X n L be a line bundle over , then we can de ne the induced bundle ∗ on by pulling back U pr1L U × X L along the projection map pr1. Therefore, we obtain a particular vector bundle 0 ∗ (2.21) E := E ⊗ pr1L −→ U × X.
6 Indeed, E0 is a twisted bundle where each ber of E0 is obtained by multiplying each ber of E by a scalar. Hence, we produce a new vector bundle E0 by "twisting" E such that E0 is not (globally) isomorphic to . Let be a local trivializing cover of such that is trivial. Then it E {Ui} U L|Ui follows from the de nition of E0 that
0 ∼ (2.22) E |Ui×X = E|Ui×X ∀i. As n is representable by a scheme , there are morphisms and in FX M fE : U → M fE0 : U → M 0 MorSch (U, M) corresponding to E and E respectively such that C
0 (2.23) fE|Ui×X = fE |Ui×X ∀i.
Since hM is a sheaf, it follows from the gluing axiom that all such morphisms are glued together nicely such that
fE|U×X = fE0 |U×X . (2.24) But, from the representability of the functor n , it implies that FX 0 ∼ E |U×X = E|U×X . (2.25)
This yields a contradiction to E E0. Remark 2.1.1. The reason behind the failure of the representability of n is that vector bundles FX have a number of non-trivial automorphisms, for instance, induced by a scalar multiplication as above. This example, in fact, may provide an important insight why a generic moduli problem is destined to be non-representable in the category of schemes. In many cases, the main source of non-representability problem turns out to be the existance of non-trivial automorphisms for the moduli problem of interest.
2.2 Moduli of elliptic curves In this section, we study the moduli space of elliptic curves, and try to show how the existence of non-trivial automorphisms again leads to the non-representability of the corresponding moduli functor. The study of such a moduli problem is indeed a classical topic, and further discussion can be found elsewhere. For an introduction, we refer to [1]. More detailed treatment (possibly with dierent approaches) can be found, for instance, in [8,5,6].
De nition 2.2.1. We rst recall that one can de ne the notion of an elliptic curve over C in a number of equivalent ways. An elliptic curve over C is de ned to be either of the following objects:
1. A Riemannian surface Σ of genus 1 with a choice of a point p ∈ Σ. 2. A quotient space where is a rank 2 lattice in for each . C/Λ Λ = ω1 · Z ⊕ ω2 · Z C ωi ∈ C 3. A smooth algebraic curve of genus 1 and degree 3 in 2 . PC We actually use the second characterization of an elliptic curve, namely the one given in terms of lattices. With this interpretation in hand, the study of the moduli of elliptic curves boils down to the study of integer lattices of full rank in C.
-action on the upper half plane and the fundamental domain. Recall that the SL2(Z) Lie group acts on the upper half plane as follows: SL2(Z) H = {z ∈ C : im(z) > 0}
a b az + b a b · z := for all ∈ SL ( ) and ∀z ∈ . c d cz + d c d 2 Z H
It is clear from the de nition of the action that both ±I act on H in the same way, and hence we will concentrate on the action of on . Then the fundamental domain PSL2(Z) := SL2(Z)/{±I} H of this action turns out to be the set Γ := H/P SL2(Z) n 1 1 o n 1 o Γ = z : − ≤ Re(z) < , |z| > 1 ∪ z : − ≤ Re(z) ≤ 0, |z| = 1 . 2 2 2
7 Figure 1: The fundamental domain Γ where µ = e2πi/3 and ρ = e2πi/6.
It is very well known that Γ is in fact a Riemann surface whose points correspond to the isomorphism classes of lattices of full rank in C up to homotheties. Note also that any lattice is isomorphic to a "normalized" lattice Λ = ω1Z ⊕ ω2Z for some (2.26) Λτ := 1 · Z ⊕ τ · Z τ ∈ H. We say that two lattices and with are homothetic if Λτ1 = 1 · Z ⊕ τ1 · Z Λτ2 = 1 · Z ⊕ τ2 · Z τi ∈ H there exits such that In other words, serves as a coarse moduli g ∈ PSL2(Z) τ2 = g·τ1. H/P SL2(Z) space for isomorphism classes of elliptic curves with . As we shall see soon, however, C/Λτ τ ∈ Γ it turns out that the space Γ is not sensitive enough to parametrize certain families of elliptic curves. This amounts to say that not all families of elliptic curves over some base B correspond to morphisms . Hence, fails to become a ne moduli space. B → H/P SL2(Z) Γ Remark 2.2.1. 1. The fundamental domain can also be represented as a free product of nite groups and Γ Z2 as follows: Z3
D 0 −1 1 1 E Γ = S := ,T := ∈ PSL ( ) | S2 = (ST )3 = I 1 0 1 0 2 Z ∼ = Z2 ? Z3.
2. The action of on is not free. Indeed, some routine computations show that PSL2(Z) H , τ = i Z2 Stab (τ) ∼ (2.27) PSL2(Z) = Z3, τ = µ or ρ {e}, else.
3. It follows from the non-freeness of the action that one has special types of lattices, namely
the square lattice Λi and the hexagonal lattice Λµ (or Λρ) such that ( Z4, τ = i Aut(Λτ ) = (2.28) Z6, τ = µ or ρ. This gives rise to non-trivial automorphisms for the corresponding elliptic curves and C/Λi by using, for instance, rotational symmetries of a square and that of a hexagon re- C/Λµ spectively. As before, the existance of non-trivial automorphisms allows us to produce some examples which eventually show that the corresponding moduli problem of elliptic curves can not be represented by the space . But, we should keep in mind that it becomes H/P SL2(Z) the coarse moduli space.
8 Moduli functor for the families of elliptic curves. We de ne the corresponding moduli functor op Fell : Sch −→ Sets, U 7→ Fell(U) (2.29) C as follows: Given a scheme U, one can de ne Fell(U) := Sell(U)/ ∼ where Sell(U) is the set of (continuous) families of elliptic curves over the base scheme U: each ber is (2.30) Sell(U) := E → U : Eu C/Λτ(u) ∀u ∈ U where is an elliptic curve with , and F . We C/Λτ(u) τ : U → H/P SL2(Z), u 7→ τ(u) E = u∈U Eu 0 say that two families πE : E → U and πE0 : E → U over U are equivalent if and only if there exists an isomorphism ∼ 0 of families such that on each ber −1 , restricts to f : E −→ E Eu = πE (u) f an automorphism of elliptic curves ∼ 0 (2.31) f|Eu : Eu −→ Eu, and the following diagram commutes:
f E E0
πE πE0
U (2.32)
Note that from the previous discussion, it is not hard to observe that the space in H/P SL2(Z) fact serves as the desired coarse moduli space for the moduli functor above. We now would like to show that the moduli functor is not representable by . Fell H/P SL2(Z) Claim: The moduli functor is not representable by . Fell H/P SL2(Z) Proof. Assume the contrary. Then, we have the following one-to-one correspondence between two sets: ∼ MorSch (U, /P SL2( )) = Fell(U). (2.33) C H Z
We rst consider a "constant" family E of elliptic curves on the interval [0, 1] where each ber Ex is of the form for all (2.34) Ex := C/Λi x. Recall that ∼ . Let be a non-trivial automorphism of given as a multiplication Aut(C/Λi) = Z4 f C/Λi by i, (2.35) f : C/Λi → C/Λi, z 7→ iz.
Then one can identify the bers E0 and E1 along the morphism f so that a particular family E of elliptic curves over S1 can be obtained. Similarly, one can construct another family E0 of elliptic curves over 1 by gluing the bers and via the identity morphism. We then obtain S E0 E1 two non-isomorphic families E and E0 of elliptic curves over S1 with the generic bers being all isomorphic. That is, [E] 6= [E0] such that
∼ 0 ∼ for all Ex = Ex = Ex x ∈ (0, 1), where and 0 denote the bers of "twisted" and "trivial" families respectively. See Figure2. Ex Ex As is representable by , there are corresponding morphisms 1 Fell H/P SL2(Z) fE , fE0 : S → Γ for E and E0 respectively such that
1 (2.36) fE , fE0 : S −→ H/P SL2(Z), s 7→ [i].
That is, each representing morphism is just the constant map. Let {Uk} be a local trivializing cover for S1 such that −1 ∼ ∼ −1 (2.37) πE (Uk) = (C/Λi) × Uk = πE0 (Uk). Then the representability condition implies that the representing morphisms are locally the same as well. That is,
0 (2.38) fE |Uk = fE |Uk ∀k.
9 Figure 2: The "trivial" and "twisted" families of elliptic curves over S1 with generically isomorphic bers.
Since h = Mor (−, /P SL ( )) is a sheaf, it follows from the gluing axiom that all H/P SL2(Z) Sch H 2 Z such morphisms are glued togetherC nicely. That is,
1 (2.39) fE = fE0 on S .
But, it follows from the representability of the functor Fell that we must have [E] = [E0], (2.40) which is a contradiction.
Remark 2.2.2. 1. The construction above shows that the correspondence 2.33 is not good enough to distinguish the "trivial" and "twisted" families of isomorphism classes of elliptic curves over S1 with generically isomorphic bers. As before, the main source of this failure is due to the existence of non-trivial automorphism group for the bers of the form The existence of non-trivial C/Λi. automorphisms, on the other hand, is due to the fact that acts on non-freely. PSL2(Z) H 2. One way of circumventing this sort of problem is to change the way of organizing the moduli data. For instance, we can use the language of stacks, and rede ne the moduli problem as a certain gruopoid-valued "functor"
F : Cop −→ Grpds (2.41)
where Grpds denotes the 2-category of groupoids with objects being categories C in which all morphisms are isomorphisms (these sorts of categories are called groupoids), 1-morphisms being functors F : C → D between groupoids, and 2-morphisms being natural transformations ψ : F ⇒ F 0 between two functors. Note that the groupoid structure, in fact, allows us to keep track of the non-trivial automorphisms as a part of the moduli data. 3. When we go back the example above, one can easily check that the space can, in H/P SL2(Z) fact, be regarded as a groupoid where objects are the elements of H, and the set of morphisms is the set In fact, PSL2(Z) × H. Mor (x, y) :=g ∈ PSL ( ): y = g · x H/P SL2(Z) 2 Z ∼ (2.42) = PSL2(Z) × H.
10 Denote a morphism by (g, x): x 7→ g · x. Note that two morphisms (g, x) and (h, y) are composable if x = h · y. Then we have
(g, h · y) ◦ (h, y) = (gh, y). (2.43)
Furthermore, the inverse of the morphism (g, x) is (g−1, g · x). Informally speaking, two non-isomorphic families E and E0 above can be represented by points like [i, f] and [i, id] via suitable constant representing morphisms as above where the second slots in the parenthesis are to keep track of possible automorphisms distinguishing these families. The last statement will be elaborated in the next section. In literature, the space is an example of H/P SL2(Z) an orbifold.
3 2-categories and Stacks
Stacks and 2-categories serve as motivating/prototype conceptual examples before introducing the notions like ∞-categories, derived schemes, higher stacks and derived stacks [2]. By using 2-categorical version of the Yoneda lemma [3], one can show that the "re ned" moduli functor
F : Cop −→ Grpds (3.1) turns out to be representable in the 2-category Stks of stacks. The price we have to pay is to adopt higher categorical dictionary, which leads to the change in the level of abstraction in a way that objects under consideration may become rather counter-intuitive. We rst brie y recall the basics of 2-category theory. For details, we refer to [7,3].
3.1 A digression on 2-categories
De nition 3.1.1. A 2-category C consists of the following data: 1. A collection of objects: Ob(C).
2. For each pair x, y of objects, a category MorC(x, y). Here, objects of the category HomC(x, y) f are called 1-morphisms and are denoted either by f : x → y or x −→ y. The morphisms of MorC(x, y) are called 2-morphisms and are denoted either by φ : f ⇒ g or
f
x φ y.
g (3.2)
The composition of two 2-morphisms α : f ⇒ g and β : g ⇒ h in MorC(x, y) is called a vertical composition, and denoted by β ◦ α : f ⇒ h or
f f
α x y, x β ◦ α y. β g
h h (3.3)
A 2-morphism α : f ⇒ g is invertible if there exits a 2-morphism β : g ⇒ f such that β ◦ α = idf and α ◦ β = idg. Furthermore, an invertible 2-morphism α : f ⇒ g is called a 2-isomorphism. It is sometimes denoted by α : f ⇔ g. 3. For each triple x, y, z of objects in C, there is a composition functor
µx,y,z : MorC(x, y) × MorC(y, z) → MorC(x, z) (3.4) which is de ned as follows:
11 (a) On 1-morphisms, it acts as the usual composition of morphisms in C:
f g g◦f µx,y,z : x −→ y, y −→ z 7−→ y −−→ z
(b) On 2-morphisms, it acts as a horizontal composition, denoted by ?:
0 0 α 0 α 0 α ?α 0 0 µx,y,z : f =⇒ f , g =⇒ g 7−→ g ◦ f ===⇒ g ◦ f (3.5) That is, given two 2-morphisms
f g
x α y, y α0 z,
0 0 f g (3.6)
0 α 0 α 0 µx,y,z maps the pair f =⇒ f , g =⇒ g of 2-morphisms to a 2-morphism
g ◦ f
x α0 ? α z.
0 0 g ◦ f (3.7)
These data must satisfy the following conditions:
For each object X of C and each 1-morphism f : A → B, we have an identity 1-morphism idX : X → X and an identity 2-morphism idf : f ⇒ f respectively. The composition of 1-morphisms (2-morphisms respectively) is associative. Horizontal and vertical compositions of 2-morphisms are "compatible" in the following sense. For a composition diagram
f g
α β x f 0 y g0 z α0 β0
00 g00 f (3.8)
we have (β0 ◦ β) ? (α0 ◦ α) = (β0 ? α0) ◦ (β ? α). (3.9)
Example 3.1.1. Every category can be realized as a 2-category. Indeed, let C be a category and MorC(A, B) denote the set of morphisms between two objects A, B. Then it is clear to observe that for any pair (A, B) of objects in C the set MorC(A, B) de nes a category MorC(A,B) whose objects (1-morphisms) are just morphisms A → B in C, and morphisms (2-morphisms) are just identities. That is, there are no non-trivial higher morphisms in this realization. A category is sometimes called a 1-category.
Remark 3.1.1. Given a 2-category C, one can obtain a category C0 by de ning Ob(C0) := Ob(C) and the "set" of morphisms in to be MorC0 (A, B) C0 (3.10) MorC0 (A, B) := MorC(A, B)/ ∼ where if there exits a 2-isomorphism . That is, is just the set of f ∼ g α : f ⇔ g MorC0 (A, B) isomorphism classes of 1-morphisms in C.
12 Example 3.1.2. A collection of categories forms a 2-category, namely the 2-category Cat of categories. Here, objects of Cat are just categories C, 1-morphisms in Cat are functors F : C → D between two categories, and 2-morphisms are natural transformations η : F ⇒ G of functors. In this example, there are no non-trivial higher n-morphisms for n > 2. Once we allow such types of morphisms, we land in the territory of higher categories. De nition 3.1.2. Let C be a 2-category. Two objects X,Y in C are said to be equivalent if there f g exist a pair (X −→ Y,Y −→ X) of 1-morphisms, and two 2-isomorphisms α : g ◦ f ⇔ idX and 0 α : f ◦ g ⇔ idY . De nition 3.1.3. A pseudo-functor F : C → D between two 2-categories C, D consists of the following data: 1. For each object A in C, an object F(A) in D,
f F(f) 2. For each 1-morphism A −→ B in C, a 1-morphism F(A) −−−→F(B) in D, 3. For each 2-morphism α : f ⇒ g in C, a 2-morphism F(α): F(f) ⇒ F(g) in D satisfying the following conditions:
(a) F respects 1- and 2-identities: F(idA) = idF(A) and F(idf ) = idF(f) for all A ∈ Ob(C) and f ∈ MorC(X,Y ). (b) F respects a composition of 1-morphisms up to a 2-isomorphism: Given a composition f g of 1-morphisms A −→ B −→ C in C, there is a 2-isomorphism F (3.11) φg,f : F(g ◦ f) ⇒ F(g) ◦ F(f) such that the following diagram commutes (encoding the associativity):
F(h ◦ g ◦ f) F(h) ◦ F(g ◦ f) F φh,g◦f F F φh◦g,f idF(h) ? φg,f F(h ◦ g) ◦ F(f) F(h) ◦ F(g) ◦ F(f) φF ? id h,g F(f) (3.12) such that φF = φF = id . f,idA idB ,f F(f) (c) F respects both vertical and horizontal compositions: Given a vertical composition β ◦ α : f ⇒ h with f
α x g y, β
h (3.13) we have F(β ◦ α) = F(β) ◦ F(α). Given a horizontal composition
g ◦ f
x α0 ? α z,
0 0 g ◦ f (3.14) with α : f ⇒ f 0 and α0 : g ⇒ g0, we have the following commutative diagram:
F(g) ◦ F(f) F(g0) ◦ F(f 0) F(α0) ? F(α) F F φg,f φg0,f 0
F(g ◦ f) F(g0 ◦ f 0) F(α0 ? α) (3.15)
13 De nition 3.1.4.
1.A prestack on C is de ned to be a particular (contravariant) pseudo-functor F : C → D where C is an ordinary category and D is the 2-category Grpds of groupoids. That is, it is a pseudo-functor F : Cop → Grpds for some category C. Recall that, in the 2-category Grpds of groupoids, objects are categories C in which all morphisms are isomorphisms (these sorts of categories are called groupoids), 1-morphisms are functors F : C → D between groupoids, and 2-morphisms are natural transformations ψ : F ⇒ F 0 between two functors.
2. Let Cτ be a site. That is, Cτ is a category C endowed with a Grothendieck topology τ. Then, a stack on Cτ is just a prestack with local-to-global properties w.r.t. τ. So, it can be considered as "a sheaf of groupoids" in a suitable sense.
3.2 2-category of Stacks and 2-Yoneda's Lemma
We like to present how stacks over a site Cτ form a 2-category. To this end, we need to introduce the notions of 1- and 2-morphisms between two stacks. De nition 3.2.1. Let C be a category, and X , Y : Cop −→ Grpds be two prestacks. A 1-morphism F : X → Y of prestacks consists of the following data:
1. For each object A in C, a functor FA : X (A) → Y(A), 2. For each morphism f : A → B in C, a 2-isomorphism
FA ◦ X (f)
X (B) Ff Y(A)
Y(f) ◦ FB (3.16) such that the following diagram commutes up to a 2-isomorphism
Ff : FA ◦ X (f) =⇒ Y(f) ◦ FB in Grpds: X (B) Y(B) FB X (f) Y(f)
X (A) Y(A) FA (3.17)
f g 3. Given a composition of 1-morphisms A −→ B −→ C in C, the corresponding 2-isomorphisms and de ne in compatible with the natural 2-isomorphisms X and Y . Indeed, Ff Fg Fg◦f φg,f φg,f using horizontal and vertical compositions of 2-morphisms, Fg◦f : FA◦X (g◦f) ⇒ Y(g◦f)◦FC is given by
Y X Fg◦f = φg,f ? idFC ◦ idY(f) ?Fg ◦ Ff ? idX (g) ◦ idFA ? φg,f . Such a 1-morphism is just a natural transformation between two pseudo-functors of 2-categories. De nition 3.2.2. Let C be a category, and F,G : X → Y be a pair of 1-morphisms of prestacks. A 2-morphism of between F and G
F
X Ψ Y
G (3.18) is a collection ΨA : FA ⇒ GA : A ∈ Ob(C) of invertible natural transformations of functors.
14 Remark 3.2.1. A 1-morphism (2-morphism respectively) of stacks is de ned as a 1-morphism (2-morphism resp.) of the underlying prestacks. Proposition 3.2.1. [3]
1. Given a category C, prestacks over C form a 2-category P reStkC of prestacks where 1- and 2-morphisms are de ned as above.
2. Stacks over a site C form a 2-category StkC of stacks with 1- and 2-morphisms being as above. Furthermore, it follows from the construction that we have the following observations [3].
Proposition 3.2.2. Let C be a category. Then the (1-) category P reShvC of presheaves (of sets) over C is a full 2-subcategory of P reStkC. In addition, if C admits a site structure, then the (1-) category ShvC of sheaves (of sets) over C is a full 2-subcategory of StkC. Remark 3.2.2. As we have already discussed in Remark 1.0.1, Yoneda's lemma 1.0.1 implies that any object X in a category C can be understood by studying all morphism into it. In other words, for any X ∈ Ob(C) one has a particular sheaf of sets op hX : C −→ Sets, Y 7−→ hX (Y ) := MorC(Y,X) (3.19) which uniquely determines X. If C admits a suitable site structure, then it follows from Proposition 3.2.2 that the sheaf MorC(·,X) can be considered as a stack with trivial 2-morphisms. We denote this stack by X. Lemma 3.2.1. (2-categorical Yoneda's Lemma for prestacks) Let Y : Cop −→ Grpds be a prestack over a category C. Then for each object X in C, there exits an equivalence of categories ∼ (3.20) Y(X) = MorP reStkC (X, Y). Proof. First, let us try to understand the objects of interests in the statement. On the left hand side, Y(X) is a groupoid, i.e. a category for which all morphisms are isomorphisms. On the right hand side, is the category of morphisms in the 2-category . 1- MorP reStkC (X, Y) P reStkC morphisms are a collection FA : X(A) → Y(A): A ∈ Ob(C) of functors with some compatibility conditions as above. Such collection is denoted by F : X → Y. On the other hand, 2-morphisms are of the form
F
X Ψ Y
G (3.21) where for each object Z, ΦZ : FZ ⇒ GZ is an invertible natural transformation. Therefore, for ∼ any f ∈ X(Z), we have an isomorphism ΨZ,f : FZ (f) −→ GZ (f) in Y(Z). To show the desired equivalence, we introduce the following functors: 1. We de ne the functor as follows: Θ: MorP reStkC (X, Y) −→ Y(X) (a) On objects (1-morphisms), F : X → Y 7−→ FX (idX ) (b) On morphisms (2-morphisms), ∼ Ψ: F ⇒ G 7−→ ΨX,idX : FX (idX ) −→ GX (idX ) 2. De ne the functor as follows: η : Y(X) −→ MorP reStkC (X, Y) (a) On objects A in Y(X), A 7−→ F (A) : X → Y. Here F (A) is given as a collection (A) of functors such that {FU : U ∈ Ob(C)} (A) f (3.22) FU : X(U) → Y(U), (U −→ X) 7−→ Y(f)(A)
where Y(f): Y(X) → Y(U). Notice that X(U) = MorC(U, X) is just a set in the rst place, but, as we remarked before, it can be viewed as a category for which all morphisms are identities. Therefore, (A) acts on morphisms of trivially. That FU X(U) is, it maps idf to id (A) . FU (f)
15 (b) η sends morphisms ϕ : A −→∼ B in Y(X) to 2-morphisms
F (A)
X Ψ(ϕ) Y
F (B) (3.23)
where (ϕ) is a collection (ϕ) (A) (B) of invertible natural Ψ ΨU : FU =⇒ FU : U ∈ Ob(C) transformations of functors where for each object f : U → X in X(U), there is an isomorphism (ϕ) (A) (B) (ϕ) ΨU (f): FU (f) −→ FU (f), ΨU (f) := Y(f)(ϕ). Notice that Y(f): Y(X) → Y(U) is a functor of groupoids, and hence it maps ϕ to an isomorphism.
3. Now, we like to show that the compositions of η and Θ are identities up to 2-isomorphisms. (a) Let A be an object in Y(X). Then we have Θ ◦ η(A) = Θ F (A) : X → Y (A) = FX (idX ) by de nition of (A) = Y(idX )(A) FX = idY(X)(A) = A since Y(idX ) is a functor of groupoids (b) Let be an object in . Then we get F : X → Y MorP reStkC (X, Y)
(FX (idX )) η ◦ Θ F = η(FX (idX )) = F (3.24)
where the 1-morphism F (FX (idX )) is de ned by, for all U ∈ Ob(C),
(FX (idX )) f FU : X(U) → Y(U), (U −→ X) 7−→ Y(f)(FX (idX )) ∼ Claim. Y(f)(FX (idX )) = FU (f). Proof. It follows directly from the de nition of a 2-morphism that for a morphism f U −→ X, there exists Ff : FU ◦ X(f) =⇒ Y(f) ◦ FX
such that the following diagram commutes up to a 2-isomorphism Ff :
X(X) Y(X) FX X(f) Y(f)
X(U) Y(U) FU (3.25) where X(f): X(X) → X(U), g 7→ g ◦ f. Therefore, we have ∼ Y(f) ◦ FX (idX ) = FU (idX ◦ f) = FU (f), (3.26) which proves the claim.
Therefore, since the claim holds for all U, we conclude that η ◦ Θ F = F (3.27) As a result, we get the desired equivalence of categories ∼ (3.28) η : Y(X) ←→ MorP reStkC (X, Y):Θ.
Remark 3.2.3. 2-categorical Yoneda's lemma implies that if Y : Cop −→ Grpds is a moduli functor for some moduli problem, then it is always representable by Y in the 2-category of (pre-)stacks.
16 References
[1] D. Ben-Zvi, Moduli spaces, The Princeton Companion to Mathematics (T. Gowers, J. Barrow- Green, I. Leader, eds.), Princeton University Press (2008). [2] G. Vezzosi, a Derived Stack?, Notices of the AMS 58 (2011) 955.
[3] F. Neumann, Algebraic stacks and moduli of vector bundles (2009). IMPA [4] R. Vakil, The Rising Sea: Foundations Of Algebraic Geometry, available at http://math.stanford.edu/ vakil/216blog/FOAGnov1817public.pdf. [5] R. Hain, Lectures on Moduli Spaces of Elliptic Curves, arXiv:0812.1803.
[6] R. Hain, Moduli of Riemann Surfaces, Transcendental Aspects, arXiv:math/0003144. [7] The Stacks Project Authors, Stacks Project, available at https://stacks.math.columbia.edu/. [8] J. Harris and I. Morrison, Moduli of Curves, Springer-Verlag(1998). [9] V. Hoskins, Moduli Problems and Geometric Invariant Theory. Lecture notes, Freie Univer- sität, Berlin (2016).
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