**28 October 2008**-
2:30 - 4:00 Michael Barr

Duality of*Z*-groups

**Abstract:**
A *Z*-group is a group that can be embedded in a cartesian power
of *Z* and a topological *Z*-group is one that can be
embedded topologically and algebraically in a power of *Z*. If
*D* and *C* denote the categories of *Z*-groups and
topological *Z*-groups, resp., then Hom(-,*Z*): *C*
→ *D* is adjoint on the left to Hom(-,*Z*): *D*
→ *C*. According to a well-known theorem of Lambek &
Rattray, the fixed categories for each composite are equivalent (dual
in this case, since the functors are contravariant). We identify the
fixed groups of *C* as the ones that are a kernel of a map
between powers of *Z* and the fixed groups of *D* as those
that have a pure embedding (meaning the cokernel is torsion-free) into
a power of *Z*.