Internal to a general category

In a category CC, for GG a group object and HGH \hookrightarrow G a subgroup object, the left/right object of cosets is the object of orbits of GG under left/right multiplication by HH.

Explicitly, the left coset space G/HG/H coequalizes the parallel morphisms

H×Gμproj GG H \times G \underoverset{\mu}{proj_G}\rightrightarrows G

where μ\mu is (the inclusion H×GG×GH\times G \hookrightarrow G\times G composed with) the group multiplication.

Simiarly, the right coset space H\GH\backslash G coequalizes the parallel morphisms

G×Hproj GμG G \times H \underoverset{proj_G}{\mu}\rightrightarrows G

Internal to Set

Specializing the above definition to the case where CC is the well-pointed topos SetSet, given an element gg of GG, its orbit gHgH is an element of G/HG/H and is called a left coset.

Using comprehension, we can write

G/H={gH|gG} G/H = \{g H | g \in G\}

Similar situation on the right.


The coset inherits the structure of a group if HH is a normal subgroup.

Unless GG is abelian, considering both left and right coset spaces provide different information.

The natural projection GG/HG\to G/H, mapping the element gg to the element gHg H, realizes GG as an HH-principal bundle over G/HG/H. We therefore have a homotopy pullback

G * G/H BH \array{ G & \to&* \\ \downarrow && \downarrow \\ G/H &\to& \mathbf{B}H }

where BH\mathbf{B}H is the delooping groupoid of HH. By the pasting law for homotopy pullbacks then we get the homotopy pullback

G/H BH * BG \array{ G/H & \to&\mathbf{B}H \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G }

Revised on October 30, 2013 23:24:47 by Colin Tan (