Normal and conormal maps in homotopy theory

Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets or the category of chain complexes over a commutative ring.Comment: 32 pages. The definition of twisting structure in Appendix B has been reformulated, leading to further slight modifications of definitions in Section 1. To appear in HH

Spaces of sections of Banach algebra bundles

Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, $X$ is a finite dimensional compact metric space, $\zeta : P \to X$ is a standard principal $G$-bundle, and $A_\zeta = \Gamma (X, P \times_G B)$ is the associated algebra of sections. We produce a spectral sequence which converges to $\pi_*(GL_o A_\zeta)$ with [E^2_{-p,q} \cong \check{H}^p(X ; \pi_q(GL_o B)).] A related spectral sequence converging to \K_{*+1}(A_\zeta) (the real or complex topological $K$-theory) allows us to conclude that if $B$ is Bott-stable, (i.e., if \pi_*(GL_o B) \to \K_{*+1}(B) is an isomorphism for all $*>0$) then so is $A_\zeta$.Comment: 15 pages. Results generalized to include both real and complex K-theory. To appear in J. K-Theor

A Note on the Non-Existence of Functors

We consider several cases of non-existence theorems for functors. For example, there are no nontrivial functors from the category of sets, (or the category of groups, or vector spaces) to any small category. See 2.3. Another kind of nonexistence is that of (co-)augmented functors. For example, every augmented functor from groups to abelian groups, is trivial, i.e. has a trivial augmentation map. Every surjective co-augmented functor from groups to perfect groups or to free groups is also trivial