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I am trying to read [this paper][1] by Lawrence Breen.

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 $Tors(X,G)$.

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 $x$. [1]: http://arxiv.org/abs/math/0611317

I am trying to read this paper by Lawrence Breen.

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 $Tors(X,G)$.

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 $x$.

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 generalisation of principal bundles?

EDIT : I am trying to understand what they mean in first paragraph. The following definintion of a principal space is standard, but note the occurance of a structural bundle of groups, rather than simply a constant one. General definition of principal bundle comes with a group $G$, a topological space $P$, a projection $\pi:P\rightarrow X$ onto space $X$, satisfying some local trivialization conditions. In this case, each fiber/structure group is homeomorphic to $G$. But, here, we do not have single group but a collection of groups one for each element of $X$ whose union they are denoting by $G$. We are in effect giving ourselves a family of groups $G_x$ parametrized by points $x\in X$, acting prinicpally on the corresponding fibers $P_x$ of $P$. I do not understand what does it mean to say acting prinicpally. I am guessing this just means that $P_x\rightarrow \{x\}$ is a principal $G_x$ bundle. Is this correct?

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

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

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.

I am trying to read this paper by Lawrence Breen.

I am trying to read [this paper][1] by Lawrence Breen.

Edit : Apologies to Lawrence Breen. 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 $x$. [1]: http://arxiv.org/abs/math/0611317