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2.4. The fibrations of Schreier and homogeneous points 25 α ∈ K[f] and c ∈ C, forces the condition in question. Conversely it is easy to
check that this condition implies that φ is a monoid homomorphism.
Proposition 2.4.4. For any monoid B, the fibers SPtB(Mon) and HPtB(Mon) contain the terminal object and are closed under finite products. Hence, the fibers SPtB(Mon) and HPtB(Mon) are unital.
Proof. The fact that SPtB(Mon) and HPtB(Mon) contain the terminal ob- ject of PtB(Mon) is Corollary 2.2.3. Let us show the closedness under finite products. Starting with two Schreier split epimorphisms (f, s) and (f′, s′) above B, the product in the fiber, which is nothing but the pullback of f along f′, as seen in Theorem 2.4.2, is consequently given by the following pullback, where s0 is the diagonal (defined by s0 (b) = (b, b)):
A ×BOO A′ //
// // A × OOA′ (2.4.2) f ×f ′ s×s′
//
B s0 B×B
Now the Schreier (resp. homogeneous) split epimorphisms are stable under products (Proposition 2.3.3), so that the right hand side vertical split epimor- phism is a Schreier (resp. homogeneous) one. And they are also stable under pullbacks, so that the left hand side split epimorphism is a Schreier (resp. homogeneous) one. The fact that the fibers in question are unital is now a consequence of the previous proposition and Proposition 1.3.2.
Proposition 2.4.5. For any monoid B, the fibers SPtB(Mon) and HPtB(Mon) are closed under equalizers as well, and thus under finite limits.
Proof. Given any parallel pair (h,h′) of morphisms in PtB(Mon):
// ′
// K[f ]
// ′ //66A
s′
consider its equalizer j in Mon; it determines a split epimorphism (φ,σ) which is the equalizer in P tB (M on). Complete the diagram by the vertical kernels; by commutation of limits, this produces the upper horizontal equalizer diagram. Since the set-theoretical Schreier retractions qf and qf′ make the upward right
66 K[φ] //
K[f] OO
K(h ) h
OO kf′ qf′
// K(j) kK(j)
//
K(h) ′
q
kf qf //
I hh
j
AOO
f sf′
σ
(( vv B
φ
h′