Page 115 - Textos de Matemática Vol. 45

9.5. Internal preorders, categories and monoids 107
An internal category X1 is a 3-truncated simplicial object: d0 d0
X3
oo s0
oo LLoo LL //
oo s0 X2 d1
d0 //
// X1 oo s0 X0
X2
// X1 X3
oo s0
// X2
(9.5.1)
s2 s1 d1
d3 d2
where the two following commutative squares are pullbacks of split epimor-
phisms:
oo s0
OO OO OO OO
d0
d2 s1 d1 s0
d0
d3 s2 d2 s1
oo s0 X1
d0

// X0
X2
oo s0
// X1
d0
The pullback on the left hand side defines X2 as the “objects of compos- able pair of arrows” and the map d1 : X2 → X1 as the “composition” map, while the pullback on the right hand side defines X3 as the “object of com- posable triples of arrows”. An internal functor is a morphism between such 3-truncated simplicial objects. An internal category is a preorder as soon as the morphisms d0 and d1 (from X1 to X0) are jointly monomorphic. It is then sufficient to have a 2-truncated simplicial object, the 3-truncation part coming for free. If E is a variety of universal algebra, then an internal preorder is just a preorder which is compatible with all the operations.
An internal monoid is an internal category such that X0 is the terminal object 1 which implies that d0 = d1 is the terminal map τX :
d0
oos0 oos0 τX //
X × X d1 // X oo 1 oo KK oo MM s0
s2 s1
d3 d2
d0
X × X × X
The map d0 : X × X → X becomes the first product projection and the map d2 the second one; the map d1 is then the internal binary operation, while the map s0 defines the unit element. The equations d1s0 = 1X = d1s1 give the unit axioms.
The monoid is commutative when, moreover, we have d1tw = d1, where the map tw : X × X → X × X is the twisting isomorphism which exchanges the projections.