Zoran Skoda
scratch

Let us work in a category CC with pullbacks; let GG be an internal group in CC.

Let ν:P×GP\nu: P\times G\to P be a right principal action and p:P×GPp:P\times G\to P the projection. Let π:PX\pi:P\to X be the coequalizer of ν\nu and pp. The principality says P×GP× XPP\times G \to P\times_X P given by (p,g)(p,pg)(p,g)\mapsto (p,p g) is an isomorphism. We do not assume PP to be trivial.

P×GpνPπX P\times G \overset{\nu}\underset{p}\rightrightarrows P \overset{\pi}\to X

We have also

P× XPp 2p 1PπX P\times_X P \overset{p_1}\underset{p_2}\rightrightarrows P \overset{\pi}\to X

where p 1,p 2p_1,p_2 make a kernel pair of π\pi. Thus the principality is equivalent to saying that ν,p\nu,p make also a kernel pair of its own coequalizer. The two diagrams above are truncations of augmented simplicial objects in CC. We want to relate these objects to monads.

For this let us suppose we work in codomain fibration. Then we have two monads in C/PC/P whose underlying functors are p !ν *p_! \nu^* and π *π !\pi^* \pi_!. The second monad is induced by a pair of adjoint functors, while the first is also easy to define. Namely to construct the component μ l\mu_l of the transformation μ:p !ν *p !ν *p !ν *\mu: p_! \nu^* p_!\nu^*\to p_!\nu^* where h:LPh: L\to P, by the universal property of the pullback there is an obvious map ν *p !ν *h\nu^* p_! \nu^* h to p *ν *hp_* \nu^* h which can be interpreted as a map p !ν *p !ν *hp *ν *hp_!\nu^* p_! \nu^* h\to p_* \nu^* h because the domains of the maps p !ν *p !ν *hp_!\nu^* p_! \nu^* h and ν *p !ν *h\nu^* p_! \nu^* h are the same by the definition and the commuting triangles can be checked easily.

The principality P×GP× XPP\times G \cong P\times_X P now induces the isomorphisms

p !ν *hπ *π !hp_! \nu^* h \cong \pi^* \pi_! h

natural in h:LPh:L\to P.

Thus the Eilenberg-Moore categories of the two monads are equivalent.

PXP\to X is an effective descent morphism with respect to codomain fibration if the comparison functor for any of the two above isomorphic monads above is an equivalence of categories.

[x^ μ,x^ ν]=i(a μx^ νa νx^ μ)[\hat{x}_\mu,\hat{x}_\nu] = i (a_\mu\hat{x}_\nu-a_\nu\hat{x}_\mu)
Layer 1 E Σ E_\Sigma E Σ out E_{\Sigma_{\text{out}}} E in E_{\text{in}} [] Σ , X ] [\Sigma, X] [] Σ out , X ] [\Sigma_{\text{out}}, X] [] Σ in , X ] [\Sigma_{\text{in}}, X] Bord ( X ) Bord(X) V V E out E_{\text{out}} in \text{in} out \text{out} exp ( S ) Σ in \exp(S_\nabla)\vert_{\Sigma_{\text{in}}} exp ( S ) \exp(S_\nabla) exp ( S ) Σ out \exp(S_\nabla)\vert_{\Sigma_{\text{out}}} E in E_{\text{in}} Layer 1 E Σ E_\Sigma E Σ out E_{\Sigma_{\text{out}}} E in E_{\text{in}} [] Σ , X ] [\Sigma, X] [] Σ out , X ] [\Sigma_{\text{out}}, X] [] Σ in , X ] [\Sigma_{\text{in}}, X] Bord ( X ) Bord(X) V V E out E_{\text{out}} in \text{in} out \text{out} exp ( S ) \exp(S_\nabla) exp ( S ) Σ out \exp(S_\nabla)\vert_{\Sigma_{\text{out}}} E in E_{\text{in}} Φ t t j j s s \Phi \union \mathcal{M} t^{t} j_j \frac{s}{s} Φ \Phi \union \mathcal{M}

Last revised on February 26, 2010 at 20:50:49. See the history of this page for a list of all contributions to it.