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Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A G-bundle is a map whose fibers have a G-action, i.e. are G-sets (if they are discrete), i.e. they are functors from G seen as a category to Sets. Likewise a C-bundle for a category C is a map whose fibers are functors from C to sets, or, if you want, a disjoint union of sets (one for each object of C) and an action by the morphisms of C - a morphism A-->B in C takes elements of the set corresponding to A to elements of the set corresponding to B.

There is a completely analogous version for topological categories also.

Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A $G$-bundle is a map whose fibers have a $G$-action, i.e. are $G$-sets (if they are discrete), i.e. they are functors from $G$ seen as a category to $\mathsf{Sets}$. Likewise a $\mathcal C$-bundle for a category $\mathcal C$ is a map whose fibers are functors from $\mathcal C$ to $\mathsf{Sets}$, or, if you want, a disjoint union of sets (one for each object of $\mathcal C$) and an action by the morphisms of $\mathcal C$ — a morphism $A \to B$ in $\mathcal C$ takes elements of the set corresponding to $A$ to elements of the set corresponding to $B$.

There is a completely analogous version for topological categories also.

Ieke Mordijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A G-bundle is a map whose fibers have a G-action, i.e. are G-sets (if they are discrete), i.e. they are functors from G seen as a category to Sets. Likewise a C-bundle for a category C is a map whose fibers are functors from C to sets, or, if you want, a disjoint union of sets (one for each object of C) and an action by the morphisms of C - a morphism A-->B in C takes elements of the set corresponding to A to elements of the set corresponding to B.

There is a completely analogous version for topological categories also.

Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A G-bundle is a map whose fibers have a G-action, i.e. are G-sets (if they are discrete), i.e. they are functors from G seen as a category to Sets. Likewise a C-bundle for a category C is a map whose fibers are functors from C to sets, or, if you want, a disjoint union of sets (one for each object of C) and an action by the morphisms of C - a morphism A-->B in C takes elements of the set corresponding to A to elements of the set corresponding to B.

There is a completely analogous version for topological categories also.

Ieke Mordijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A G-bundle is a map whose fibers have a G-action, i.e. are G-sets (if they are discrete), i.e. they are functors from G seen as a category to Sets. Likewise a C-bundle for a category C is a map whose fibers are functors from C to sets, or if you want a disjoint union of sets (one for each object of C) and an action by the morphisms of C - a morphisms A-->B in C takes elements of the set corresponding to A to elements of the set corresponding to B.

There is a completely analogous version for topological categories also.

Ieke Mordijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A G-bundle is a map whose fibers have a G-action, i.e. are G-sets (if they are discrete), i.e. they are functors from G seen as a category to Sets. Likewise a C-bundle for a category C is a map whose fibers are functors from C to sets, or, if you want, a disjoint union of sets (one for each object of C) and an action by the morphisms of C - a morphism A-->B in C takes elements of the set corresponding to A to elements of the set corresponding to B.

There is a completely analogous version for topological categories also.