In algebraic topology, we study invariants of spaces
with values in algebraic objects.
Homology groups
and
homotopy groups
are invariants with values in
Abelian groups
. On the other hand,
fundamental groups
take values in notnecessarily Abelian groups. Groups
are also used to construct spaces. For example,
important spaces can be constructed as
classifying spaces
of groups.
Conversely, geometric techniques are useful when we
study inﬁnite groups.
It should be noted that
categorical viewpoints
are useful when we use groups. For example, a group
is a
small category
with a single object in which all morphisms are
invertible. Classifying spaces are constructed for
groups ﬁrst. And then the construction was
extended to small categories. Furthermore group
theoretic concepts such as center and semidirect
product can be extended to categories. I don’t
know who ﬁrst found a generalization of center
to categories but it can be found in
Chirvasitu’s
[
Chi
]
. A categorical generalization of semidirect product
is the
Grothendieck construction
.
Another categorical viewpoint is, although
tautological, to regard a group as a group object in
the category of sets. As a generalization, we consider
group objects in other
monoidal categories
. For example,
[
BW
]
by Blohmann and Weinstein.
