Notion of Torsors

Question

I am trying to read this paper by Lawrence Breen.

It starts with the definition of a torsor.

Let $G$ be a bundle of groups on a space $X$. The following definition of a principal space is standard, but note the occurrence of structural bundle of groups,rather than simple a constant one. We are in effect giving ourselves a family of groups $G_x$, parametrised by points $x\in X$, acting principally on the corresponding fibers $P_x$ of $P$.

Definition: A left principal $G$-bundle (or left $G$-torsor) on a topological space $X$ is a space $\pi:P\rightarrow X$ above $X$, together with a left group action $G\times_XP\rightarrow P$ such that the induced morphism $G\times_XP\rightarrow P\times_XP$ given by $(g,p)\mapsto (gp,p)$ is an isomorphism. We require in addition that there exists a family of local sections $s_i:U_i\rightarrow P$, for some open cover $\{U_i\}$ of $U$. The groupoid of left $G$-torsors on $X$ will be denoted by $Tors(X,G)$.

I do not understand: what does it mean to say a bundle of groups on a space? Does it mean that $G$ as a set is a disjoint union of groups indexed by elements of $X$?

I do not understand anything that is said in the first paragraph.

I do not understand the necessity of defining (considering) action of $G$ on $P$ as map $G\times_XP\rightarrow P$ and not just some continuous map $G\times P \rightarrow P$.

I realise reading some other (equivalent) definition of torsors that they are just generalisations of principal $G$-bundles for some group $G$. Given a principal $G$-bundle $\pi:P\rightarrow M$ there is an open cover $\{U_i\}$ of $M$ and collection of local trivialisations $\phi_i:\pi^{-1}(U_i)\rightarrow U_i\times G$. This would give sections $s_i:U_i\rightarrow P$ given by $x\mapsto \phi_i^{-1}(x,e)$. I am guessing this is generalised to existence of sections in the definition of torsors (correct me if I am wrong).

Suppose we have a Lie group $G$ that acts on a manifold $M$. Then, we have a quotient space $M/G$ which does not have a smooth manifold structure in general. Then, we impose (with other mild conditions) that $G$ acts properly on $M$ i.e, the map $G\times M\rightarrow M\times M$ given by $(g,m)\mapsto (gm,m)$ is a proper map. Which will then confirm that the quotient space is a manifold and the obvious map $M/G\rightarrow M$ is a first example of a principal bundle.

In the definition of torsors they are asking for a similar map to be an isomorphism. I am guessing the above paragraph is the motivation for asking for such a condition to hold (correct me if I am wrong).

My question is: is my understanding (as of now) of torsors correct? Is there anything else that I am missing? What are they really other than being a generalization of principal bundles?

On a lighter note, why can't authors specify everything clearly? Or Is this how one usually write a paper?

Edit : I saw just now that he does say what he mean by bundle of groups.

Since the concepts discussed here are very general, we have at times not made explicit the mathematical objects to which they apply. For example, when we refer to “a space” this might mean a topological space, but also “a scheme” when one prefers to work in algebro-geometric context, or even “a sheaf” and we place ourselves implicitly in the category of such spaces, schemes, or shaves. Similarly, the standard notion of an $X$-group scheme $G$ will correspond in a topological context to that of a bundle of groups on a a space $X$. By this we mean a total space $G$ above space $X$ that is a group in the cartesian monoidal category of spades over $X$. In particular, the fibers $G_x$ of $G$ at points $x\in X$ are topological groups, whose group laws vary continuously with $x$.

Answer 1

I do not understand what does it mean to say bundle of groups on a space? Does it mean that G as a set is disjoint union on groups indexed bybelements of X?

Presumably there's also a condition that the groups "vary continuously" in some sense, just like a vector bundle isn't any disjoint union of arbitrary vector spaces indexed by elements of a space.

The paper seems to give an example in example 1.3.

I do not understand the necessity of defining(considering) action of G on P  as map $G \times_X P \to P$

Well as you noted before G is a disjoint union of groups, so it's not going to be a group itself and therefore can't act on things. Instead, you want individual actions of $G_x$ on $P_x$ which "vary continuously" in some sense, and that's what a fiber product gives you.

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For general background on torsors, John Baez has a nice writeup in This Week's Finds which you can find with Google. (I'm writing on mobile.)

Answer 2

It is interesting to start with an example, consider for example $\mathbb{R}^n$ on which acts simply and transitively the nilpotent Lie group $N$. Let $X$ be the space reduced to the point $\{x\}$, You have here a $N\times \{x\}$ and the bundle $P=\{x\}\times \mathbb{R}^n$ over $x$, which is a $\{x\}\times N$ torsor defined by the action above.

Then consider a principal $G$-bundle $E$ over $X$, and $P\rightarrow X$ a fibre bundle over $X$ such that $G$ acts simply and transitively on the fibre $F$, suppose that $(U_i)_{i\in I}$ is a trivialisation of $E$ and $G$, $P$ is a torsor if the maps

$U_i\times G\times F\rightarrow X\times F\times F$ by $(x,g,y)=(x,gy,y)$ can be extended to $E\times_XP\rightarrow P\times_XP$.

The general situation is when $p_E:E\rightarrow X$ is not a principal bundle, but a fibre bundle such that the fibre $G_x$ is a Lie group, and $p_P:P\rightarrow X$ such that the fibre $P_x$ of $x$ is endowed with a simply transitive action of $G_x$; $E\times_XP=\{(e,p), e\in E, p\in P:p_E(e)=p_P(p)\}$ it is the union of $G_x\times P_x$, the condition means that $f_x:G_x\times P_x\rightarrow P_x\times P_x$ defined by $f_x(g,p)=(gp,p)$ is an isomorphism.

Answer 3

I have found the notion of group bundle in Alexandre Grothendieck's "A general theory of fibre spaces with structure sheaf".

This is for my own reference and may be useful for some one who reads that article Notes on $1$- and $2$-gerbes.

We need some definitions to go to the definition of group bundle and action $G\times _XP\rightarrow P$.

Definition : A fibre space over a space $X$ is a triple $(X,E,p)$ of the space $X$, a space $E$ and a continuous map $p$ of $E$ into $X$. A homomorphism of a fibre space $(X,E,p)$ to another fibre space $(X',E',p')$ is a pair of maps $f:X\rightarrow X'$ and $g:E\rightarrow E'$ such that $p'\circ g=f\circ p$. In this case $g$ maps fibres to fibres.

Definition : Let $(X,E,p)$ be a fibre space over $X$. Let $f:X'\rightarrow X$ be a continuous map. Then we define inverse image of fibre space $(X,E,p)$ to be the fibre space $(X',E',p')$ where $E'=\{(a,e):f(a)=p(e)\}\subseteq X'\times E$ and $p':E'\rightarrow X'$ is given by $p'(a,e)=a$.

Definition : Let $(X,E,p)$ be a fibre space and $(X,E',p')$ be another fibre space. This gives a map $$p\times p': E\times E'\rightarrow X\times X$$ with $(a,b)\mapsto (p(a),p'(b))$ making $(X\times X, E\times E', p\times p')$ into a fibre space. Consider diagonal map $\Delta:X\rightarrow X\times X$ given by $x\mapsto (x,x)$. Inverse image of $(X\times X, E\times E', p\times p')$ under $\Delta$ is what is called as fibre product of $(X,E,p)$ with $(X,E',p')$, denoted by $E\times_X E'$.

Definition : Let $E$ be a fibre space over $X$, provided with the supplement structure defined by a homomorphism of the fibre product $E\times_X E\rightarrow E$, or what is the same, a law of composition defined in each fibre $E_x$ such that the corresponding global map $E\times_X E\rightarrow E$ be continuous. This is called fibre space with composition law.

Definition : A group bundle $E$ over $X$ is a fibre space with composition law over $X$ such that for each $x\in X$, the fibre $E_x$ of $E$ is a group, the unit of which depends continuously on $x$, and that the map of $E$ into itself which on each fibre $E_x$ reduces to $z\mapsto z^{-1}$ be continuous.

Definition : Let $G$ be a group bundle on $X$ and $A$ be a fibre space on $X$. We say that $G$ operates at left on $A$ if we are given a homomorphism $G\times_X A\rightarrow A$ such that for each $x\in X$ the corresponding map $G_x\times A_x\rightarrow A_x$ is a group action.

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I am almost sure that by the unit of which depends continuously on $x$ it means the map $X\rightarrow E$ given by $x\mapsto e_x\in E_x\subseteq E$ is a continuous map, correct me if I am wrong.

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In case of that article, $G$ is a bundle of groups on $X$ (i.e., fibre space $G\rightarrow X$) and $P\rightarrow X$ is a fibre space. Then there is fibre prodcut $(G\times_X P,X)$.

By $G\times_X P\rightarrow P$ it actually means homomorphism of fibre space $(G\times_X P,X)$ to $(P,X)$ i.e., it maps fibres to fibres in particular $G_x\times P_x$ is mapped to $P_x$.