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108 Chapter 9. Appendix
Example 9.5.1. If E is the category Top of topological spaces and continuous maps, an internal monoid in E is what is usually called a topological monoid, i.e. a topological space X endowed with a monoid operation ·: X × X → X which is continuous.
Example 9.5.2. If E is the category M on of monoids, an internal monoid in E is nothing but a commutative monoid. Indeed, the monoid operation ·: X×X → X of a monoid X is a homomorphism of monoids if and only if X is commutative. This is the classical Eckmann-Hilton argument [18].
When we are working in a (pointed) unital category C, the cooperator m: A×A → A of a commutative object satisfies by definition the unit axioms. The commutativity mtw = m of the binary operation can be checked using the fact that the pair (⟨1A , 0⟩, ⟨0, 1A ⟩) is jointly strongly epimorphic, while the associativity can be checked using the fact that the pair (⟨1A,0⟩, ⟨0,1A×A⟩) is jointly strongly epimorphic. This is why a commutative object in a unital category is endowed with a canonical structure of an internal commutative monoid. The example of the internal monoids in M on, that are the commutative monoids, is just an instance of this fact.
9.6 Internal groupoids, equivalence relations, groups
An internal category X1 in E is a groupoid when, in addition, the following square determined by the composition map d1 is a pullback in E, see [7]:
X d1 // X (9.6.2) 21
d0 X1
d0 X0,
// d0
Actually it appears to be only a 3-truncated simplicial object:
X3
d0 d0 oos0 oo s0
d0 //
X2 d1 // X1 oo s0 X0
ooLLoo
s2 s1
LL // d1
d3 d2