Crossed Modules as Homotopy Normal Maps
نویسندگان
چکیده
In this note we consider crossed modules of groups (N → G, G→ Aut(N)), as a homotopy version of the inclusion N ⊂ G of a normal subgroup. Our main observation is a characterization of the underlying map N → G of a crossed module, in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in particular we consider briefly what we call ‘normal maps’ between simplicial groups.
منابع مشابه
Crossed Modules Induced by an Inclusionof a Normal Subgroup , with Applications
We obtain some explicit calculations of crossed Q-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups.
متن کاملComputing Crossed Modules Induced by an Inclusion of a Normal Subgroup with Applications to Homotopy Types
We obtain some explicit calculations of crossed Q modules induced from a crossed module over a normal subgroup P of Q By virtue of theorems of Brown and Higgins this enables the computation of the homotopy types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups Introduction A crossed module M M P has a classifying space BM see for example ...
متن کاملComputing Crossed Modules Induced by an Inclusion of a Normal Subgroup, with Applications to Homotopy 2-types
We obtain some explicit calculations of crossedQ-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups. Introduction A crossed module M = (μ : M → P ) has a classifying space BM (...
متن کاملNormal 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 mon...
متن کاملar X iv : m at h / 06 04 02 9 v 2 [ m at h . A T ] 1 7 O ct 2 00 6 SECONDARY HOMOTOPY GROUPS
Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a 2-functor on the groupoid-enriched category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n− 1)-connected (n+1)-types for n ≥ 0. Introduction The computation of homotopy groups of spheres in low degrees in [Tod62] uses heavily secon...
متن کامل