Definition: C is called connected if it has exactly one component, i.e. FC = 1 (where F is the left adjoint to the
inclusion I of the category of discrete objects). This is equivalent
to pi0◦C = 1, where pi0 = IF.
Definition: A group is a monoid in which every element has an inverse.
The group case of monoid has special features which arise from the following:
- Each congruence relation on a group arises from a normal subgroup
- for any two representations, the underlying set of the map space is the map set of the underlying sets
- given elements x and y in a connected representation, there is a group element g with x◦g = y.
Theorem: The number of non-isomorphic connected representations of a group G is
at most the number of subgroups of G.
Constants and codiscrete objects
Definition: A constant of a monoid M is an element c such that cm = c for all elements m of M.
The codiscrete inclusion J is right-adjoint to the fixed-point functor G. Indeed, we have a string of four adjoints F -| I -| G -| J relating the M-actions to the abstract set. While FI ≈ GI ≈ GJ are all equivalent to the identity functor on S, the composite FJ assings, to every set S, the set of (tokens for) connected components of the codiscrete action on S[C].
For S =/= 0, FJS = 1.
Monoids with at least two constants
Proposition: If a monoid M has at least two constants, then for every nonempty set S, the codiscrete action
J(S) is connected.
In fact, there are more connected M-actions. If M has at least two constants, then the truth-value space Omega in the category of right M-actions is connected. For any right M-action X, the map space Omega[X] is connected. Any X is the domain of a subobject of a connected object, for example X -> Omega[X] by the ‘singleton’ inclusion.
