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 categoryof 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 $x$.