By Robert Gordon
A $G$-category is a class on which a gaggle $G$ acts. This paintings experiences the $2$-category $G$-Cat of $G$-categories, $G$-functors (functors which shuttle with the motion of $G$) and $G$-natural modifications (natural ameliorations which travel with the $G$-action). there's specific emphasis at the courting among a $G$-category and its solid subcategory, the most important sub-$G$-category on which $G$ operates trivially. additionally contained listed here are a few very common functions of the speculation to numerous additive $G$-categories and to $G$-topoi.
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Additional resources for G-Categories
Thus, provided M can be defined on arrows so as to be a G-functor, X is a 2-cell. But, inasmuch as (MP,Ap) = (MR,AR) is initial, Mx must be defined for x e P(P,P l) by commutativity of i NMP- -*• P ' |x>' NMx •> NMP' R. 38 GORDON But then, it follows that -» y P' g NMx NgMx ->N 9 MP »^ NMV NMgx commutes; and so gMx = M g x. 1, all that is needed to finish is as in the usual pointwise construction of adjoints: p defined by commutativity of N AN NMN > Np N makes, by naturality of A -> NM ANN! v NM commute; so pM • MA = 1.
Ii) C(C,-) is trivially transversally representable whenever C is 46 R. GORDON transversaled . (iii) C is stably closed. Proof, (i)^-(iii). (iii)^(ii). 4. Suppose C is transversaled by <$>. 2, $ has a limit,
2. The stable diagonal A: G-Set -» stab(par(G-Set)) G is an equivalence with left adjoint-left inverse the canonical functor sending systems of isomorphisms to the G-sets they induce, We say that a G-category is hereditarily complete H-category for every stably subgroup H of G. 3. Par(G-Set) is hereditarily stably complete and cocomplete. particular, it is hereditarily stably closed. ,W,) = (X ,W ) for all geG, then the cartesian product of the sets X. ) under the rule (gx) j J = g x -1, J 9j all x e nx..
G-Categories by Robert Gordon