G-Categories by Robert Gordon

By Robert Gordon

A $G$-category is a class on which a gaggle $G$ acts. This paintings stories the $2$-category $G$-Cat of $G$-categories, $G$-functors (functors which travel with the motion of $G$) and $G$-natural adjustments (natural variations which travel with the $G$-action). there's specific emphasis at the courting among a $G$-category and its sturdy subcategory, the most important sub-$G$-category on which $G$ operates trivially. additionally contained listed here are a few very basic purposes of the speculation to varied additive $G$-categories and to $G$-topoi.

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Obviously, FT = 0. NFZ From this we see that (Nq)p' = p, so that q is an arrow (Q',p') -» (Q,p) in (P+N). By faithfulness of F, q is in fact the unique such arrow satisfying T • q = T'. Thus T is a limiting H-cone. Now, above, when F T ' = 0 one gets q = L ; so T 1 = T in that case. creates an H-limit for Z. n Thus F R. 6 more difficult to apply than the ordinary adjoint functor theorem. 2, whereby a G-functor, upon having a left adjoint, has a left G-adjoint. 9. 9. A cosystem of isomorphisms cj> in (P+N) is costabilizing pre- cisely when the G(P)-cosystem of isomorphisms in Q.

G h g n= T'T^ • T'Th- 9T'T. • 9T. T = T'T^ • J ' \ g g ,h = T'(x . x. )y • v(x g h g h h g h h • gx' T • V hT 9 • x')T = T'x . • x« T =v (x»x) . ; h' gh gh gh and, consequently, the composition (2) is defined whenever T'T is defined. With this composition, we form the category Trans-Cat of G-categories and transversaled functors. 32 R. GORDON Now, if n: (M,y) •* (N,v): A -»- 8 is a transverse natural transformation and (T,T): 9 B + C and (S,a): V + A are transversaled functors, the diagrams TM I xgM 9 TM M TM9 gT n Tgn 9 J nS 9 MS >9TN V -*9NS V ;< j£s—^Ngs g MS -^ T N 4 Ma Na MS9 TTI9 nS9 ^NSg are commutative, because the upper square on the left commutes by naturality of T and the lower square on the right commutes by naturality of n.

In particular, if C is transversaled by <)> and n is a transversaled representation of T by C c r then n is a representation of T by (C,); and if C is trivially transversaled P and n a trivially transversaled representation of T by C, triv- n is a reprep sentation of triv- T by C.

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