In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object.
Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks.
Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category C; for example, C can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology.
Let F be a category and suppose it is fibered over C through the functor
p:F\toC
Given an object U in C and objects x, y in
F(U)=p-1(U)
f:V\toU
f*x,f*y
\underline{\operatorname{Hom}}(x,y)(V\overset{f}\toU)=[\operatorname{Hom}(f*x,f*y)]
f*x
f*y
g:W\toV
\underline{\operatorname{Hom}}(x,y)(V\overset{f}\toU)\to\underline{\operatorname{Hom}}(x,y)(W\overset{f\circg}\toU)
[\operatorname{Hom}(f*x,f*y)]\overset{g*}\to[\operatorname{Hom}(g*(f*x),g*(f*y))]=[\operatorname{Hom}((f\circg)*x,(f\circg)*y)]
g*\circf*\simeq(f\circg)*
\underline{\operatorname{Hom}}(x,y)
C/U
By definition, F is a prestack if, for each pair x, y,
\underline{\operatorname{Hom}}(x,y)
C/U
This definition can be equivalently phrased as follows. First, for each covering family
\{Vi\toU\}
F(\{Vi\toU\})
p1:Vi x UVj\toVi,p12:Vi x UVj x UVk\toVi x UVj
\{(xi,\varphiij)\}
xi
F(Vi)
\varphiij:
* | |
p | |
2 |
xj\overset{\sim}\to
* | |
p | |
1 |
xi
* | |
p | |
13 |
\varphiik=
* | |
p | |
12 |
\varphiij\circ
* | |
p | |
23 |
\varphijk
\{(xi,\varphiij)\}\to\{(yi,\psiij)\}
\alphai:xi\toyi
F(Vi)
\psiij\circ
* | |
p | |
2 |
\alphaj=
* | |
p | |
1 |
\alphai\circ\varphiij.
There is an obvious functor
F(U)\toF(\{Vi\toU\})
\{Vi\toU\}
F(U)\toF(\{Vi\toU\})
The essential image of
F(U)\toF(\{Vi\toU\})
\{Vi\toU\}
F(U)\toF(\{Vi\toU\})
These reformulations of the definitions of prestacks and stacks make intuitive meanings of those concepts very explicit: (1) "fibered category" means one can construct a pullback (2) "prestack in groupoids" additionally means "locally isomorphic" implies "isomorphic" (3) "stack in groupoids" means, in addition to the previous properties, a global object can be constructed from local data subject to cocycle conditions. All these work up to canonical isomorphisms.
See also: Morphism of algebraic stacks.
Given prestacks
p:F\toC,q:G\toC
f:F\toG
q\circf=p
If
p:FS\toC
p-1(U)=FS(U)
FU
\operatorname{Funct}C(U,F)\overset{\chi\mapsto\chi(1U)}\toF(U)
\operatorname{Funct}C
Let
f:F\toB,g:G\toB
F x B,G=F x BG
(x,y,\psi)
\psi:f(x)\overset{\sim}\tog(y)
(x,y,\psi)\to(x',y',\psi')
\alpha:x\tox'
\beta:y\toy'
g(\beta)\circ\psi=\psi'\circf(\alpha)
F x BG
This fiber product behaves like a usual fiber product but up to natural isomorphisms. The meaning of this is the following. Firstly, the obvious square does not commute; instead, for each object
(x,y,\psi)
F x BG
\psi:(f\circp)(x,y,\psi)=f(x)\overset{\sim}\tog(y)=(g\circq)(x,y,\psi)
\Psi:f\circp\overset{\sim}\tog\circq
u:H\toF
v:H\toG
f\circu\overset{\sim}\tog\circv
w:H\toF x BG
u\overset{\sim}\top\circw
q\circw\overset{\sim}\tov
f\circu\overset{\sim}\tog\circv
f\circp\circw\overset{\sim}\tog\circq\circw
F x BG
When B is the base category C (the prestack over itself), B is dropped and one simply writes
F x G
\psi
Example: For each prestack
p:X\toC
\Delta:X\toX x X
x\mapsto(x,x,1p(x))
Example: Given
Fi\toBi,Gi\toBi,i=1,2
(F1 x F2)
x | |
B1 x B2 |
(G1 x G2)\simeq(F1
x | |
B1 |
G1) x (F2
x | |
B2 |
G2)
Example: Given
f:F\toB,g:G\toB
\Delta:B\toB x B
F x BG\simeq(F x G) x BB
A morphism of prestacks
f:X\toY
S\toY
X x YS
In particular, the definition applies to the structure map
p:X\toC
X\simeqX x CC
The definition applies also to the diagonal morphism
\Delta:X\toX x X
\Delta
U\toX
U x XT\simeq(U x T) x XX
If
f:X\toY
S\toY
X x YS\toS
f:X\toY
T\toY
X x YT\toT
Let G be an algebraic group acting from the right on a scheme X of finite type over a field k. Then the group action of G on X determines a prestack (but not a stack) over the category C of k-schemes, as follows. Let F be the category where
(U,x)
X(U)=\operatorname{Hom}C(U,X)
(U,x)\to(V,y)
U\toV
g\inG(U)
y':U\toV\overset{y}\toX
Through the forgetful functor to C, this category F is fibered in groupoids and is known as an action groupoid or a transformation groupoid. It may also be called the quotient prestack of X by G and be denoted as
[X/G]pre
[X/G]
When X is a point
*=\operatorname{Spec}(k)
[*/G]pre=BGpre
One viewing X as a prestack (in fact a stack), there is the obvious canonical map
\pi:X\toF
(U,x:U\toX)
(U,x)\to(V,y)
U\toV\overset{y}\toX
Then the above canonical map fits into a 2-coequalizer (a 2-quotient):
X x G\overset{s}\underset{t}\rightrightarrowsX\overset{\pi}\toF
\pi\circs=\pi\circt
\pi\circs\overset{\sim}\to\pi\circt
g:(\pi\circs)(x,g)=\pi(x)\overset{\sim}\to(\pi\circt)(x,g)=\pi(xg).
Let X be a scheme in the base category C. By definition, an equivalence pre-relation is a morphism
R\toX x X
f(T):R(T)=\operatorname{Hom}(T,R)\toX(T) x X(T)
f(T)
Example: Let an algebraic group G act on a scheme X of finite type over a field k. Take
R=X x kG
f(T):R(T)\toX(T) x X(T),(x,g)\mapsto(x,xg).
To each given equivalence pre-relation
f:R\toX x X
s=p1\circf,t=p2\circf
f:V\toU
C/U
\begin{align} \underline{\operatorname{Hom}}(x,y)(V\overset{f}\toU)&=[\operatorname{Hom}(f*x,f*y)]\\ &=[\{\delta:V\toR|s\circ\delta=f*x,t\circ\delta=f*y\}]\\ &=[\{\delta:V\toR|(s,t)\circ\delta=(x,y)\circf\}]. \end{align}
\underline{\operatorname{Hom}}(x,y)
(s,t):R\toX x X
(x,y):U\toX x X
\underline{\operatorname{Hom}}(x,y)
The prestack F above may be written as
[X/\simR]pre
[X/\simR]
Note, when X is viewed as a stack, both X and
[X/\simR]pre
[X/\simR]pre
\delta
One importance of this construction is that it provides an atlas for an algebraic space: every algebraic space is of the form
[U/\simR]
f:R\toU x U
f(T):R(T)\toU(T) x U(T)
s,t:R\toU x U\toU
ak{X}
f:R\toU x U
ak{X}
ak{X}\simeq[U/\simR]
\pi:U\toak{X}
U x ak{X
s,t:R\rightrightarrowsU
f=(s,t):R\toU x U
f
\pi:U\toak{X}
\pi:[U/\simR]pre\toak{X}
\delta
\pi
[U/\simR]\toak{X}
There is a way to associate a stack to a given prestack. It is similar to the sheafification of a presheaf and is called stackification. The idea of the construction is quite simple: given a prestack
p:F\toC
As it turns out, it is a stack and comes with a natural morphism
\theta:F\toHF
In some special cases, the stackification can be described in terms of torsors for affine group schemes or the generalizations. In fact, according to this point of view, a stack in groupoids is nothing but a category of torsors, and a prestack a category of trivial torsors, which are local models of torsors.