The next step is to specify a choice of factorisations for the diagonal maps δf. With these in tow we can
mimic the interpretation of identity types sketched at the end of the previous chapter. We will require certain regularity principles for these choices in order for substitution to be modelled soundly, and to express these we need to fit our clovenLandR-maps into their own categories.
Definition 5.5 (Morphism Of Cloven Maps). Given cloven L-maps (f, s) : X →Y and (g, t) : Z → W a morphism of clovenL-maps (h, k) : (f, s)→(g, t) is given by a commutative square
X Z
Y W
h
f g
k
with the additional property that the following square commutes
Y P f
W P g
s
k P(h,k)
t
Dually, given cloven R-maps (f, p) : X → Y and (g, r) : Z → W a morphism (h, k) : (f, p) → (g, r) is a commutative square as above, with the additional property that the following square commutes
P f P g
X W
P(h,k)
p r
h
This notion allows us to prove a number of properties that we give in the following lemma.
Lemma 5.6.
1. Given a cloven weak factorisation system on a categoryC there exist a categoryL-MapC comprised of cloven L-maps and morphisms of clovenL-maps and a categoryR-MapC comprised of clovenR-maps and morphism of clovenR-maps.
2. Precomposition by a morphism of clovenL-maps sends canonical fillers to canonical fillers; postcompo- sition by a morphism of cloven R-maps sends canonical fillers to canonical fillers.
3. The morphismp[f]of (5.4) is the unique such clovenR-map structure onpturning the pullback square into a morphism of clovenR-maps.
Proof. 1. By definingid(f,p)= (iddom(f), idcod(f)) and (h0, k0)◦(h, k) = (h0◦h, k0◦k) we obtain categories L- MapC andR-MapC. We have the commutative composite:
X Z Z0 Y W W0 h f g h0 g0 k k0
Then in the case ofL-maps the commutative composite Y P f W P g W0 P g0 s k P(h,k) t k0 P(h0,k0) t0
and in the case ofR-maps the commutative composite
P f P g P g0 X W W0 P(h,k) p P(h0,k0) r g0 h h0
The functorality of the assignmentP(h, k) thus completes the proof that this notion of composition is well defined.
2. Suppose we have a cloven L-map (f, s) : U →V and anR-map (g, p) : X →Y. As before, given a commutative square U X V Y h f g k
we have a canonical choice of filler given by p◦P(h, k)◦s. Supposing we have a morphism of cloven
L-maps (i, j) : (f0, s0)→(f, s) we obtain the commutative composite:
U0 X
V0 Y
hi
f0 g
kj
We then have an obvious choice of filler given by precomposing the canonical choice for the original diagramp◦P(h, k)◦swith the mapj. By the defining property of the cloven morphism it follows that
p◦P(h, k)◦s◦j=p◦P(h, k)◦P(i, j)◦s0 =p◦P(hi, kj)◦s0
That is, the canonical choice of filler for the composite diagram. This works analogously for morphisms of cloven R-maps and post-composition.
3. Finally we note that by commutativity of (5.4) (f+, f) : (π
A[f], p[f])→(πA, p) is a morphism of cloven
R-maps. Not only this, but by the universal property of the pullbackp[f] is theunique clovenR-map structure on πA[f] such that (f+, f) is a morphism ofR-maps. We collect the preceding discussion in
a lemma.
Taken together, this shows that this notion of morphism is a good one, and equipped with the data given thus far we have tight control over our canonical choice of fillers. With one more property we will be able to prove that our type category has identity types. Recall that given a morphismf :X →Y we obtain the diagonal mapδf from the following pullback
X X×Y X X X Y idX idX δf f f
Definition 5.7 (Choice Of Diagonal Factorisations). A cloven weak factorisation system has a choice of diagonal factorisationsif, for every clovenR-map (f, p) :X→Y ofCwe have an assignment of a factorisation ofδf:
X I(f) X×Y X
if jf
together with an assignment of a cloven L-map structure on if and a cloven R-map structure on jf. This
choice isfunctorial if this assignment can be extended to a functorI:R-MapE →R-MapE×EL-MapE. That is, for every morphism of clovenR-maps (h, k) : (f, p)→(g, r) there is given a mapI(h, k) functorial inh, k such that the following squares commute:
X U I(f) I(g) h if ig I(h,k) I(f) I(g) X×Y X U×V U I(h,k) jf jg h×kh (5.5)
with the left hand square a morphism of cloven L-maps and the right hand square a morphism of cloven
R-maps. Finally the choice isstable if the square
I(f) I(g)
X×Y X Z×W Z
I(h,k)
jf jg
h×kh
Theorem 5.8. Given a cloven weak factorisation system for a categoryC, if there exists a functorial and stable choice of diagonal factorisations then the associated type category of Theorem5.4 has identity types.
Proof. We show each property can be satisfied case by case.
• Formation/Introduction: Let A ∈ T y(Γ) be given by a cloven R-map (πA, p) : Γ.A → Γ. By the
substitution property of the type category we have that the following square is a pullback:
Γ.A.A+ Γ.A
Γ.A Γ
(πA)+
πA+ πA
πA
We thus obtain the diagonal mapδπA: Γ.A→Γ.A.A
+ and use our choice of diagonal factorisation to
obtain
Γ.A I(πA) Γ.A.A+
iπA jπA
together with an assignedsandpmaking (iπA, s) a clovenL-map and (jπA, p) a clovenR-map. We thus
take (jπA, p) to be the identity typeIdA∈T y(Γ.A.A
+), makingI(π
A) = Γ.A.A+.IdA andjπA =πIdA.
We can then takeiπAas the introduction morphismrA: Γ.A→Γ.A.A +.Id
A, and clearlyπIdA◦rA=δπA
as required.
• Elimination/Computation: SupposeC∈T y(Γ.A.A+.Id
A) and we have a commutative square
Γ.A Γ.A.A+.IdA.C
Γ.A.A+.Id
A Γ.A.A+.IdA
d
rA πC
id
Since πC has a clovenR-map structurepandrAhas a clovenL- map structures, by Proposition5.3
we have a canonical choice of fillerJ(C, d) =p◦P(d, id)◦s: Γ.A.A+.Id
A→Γ.A.A+.IdA.C.
• Substitution:
1. IdA[f++] =IdA[f]: We first note that Lemma5.6iii), the pullback square
∆.A[f] Γ.A ∆ Γ f+ πA[f] πA f (5.6)
is a morphism of cloven R-maps when πA andπA[f] are equipped with their assigned clovenR-
square
∆.A[f].A[f]+.Id
A[f] Γ.A.A+.IdA
∆.A[f].A[f]+ Γ.A.A+
I(f+,f)
πIdA[f] πIdA
f++
underlying a morphism ofR-maps. By stability this square is a pullback and soI(f+, f) =f+++ . We then note that since the assigned R-map structure on πA[f] is the unique such structure
making the square a morphism of R-maps it follows that it is identical to the assigned R-map structure forπIdA[f++] and hence the required identity holds.
2. Commutativity of
∆.A[f] ∆.A[f].A[f]+.Id
A[f]
Γ.A Γ.A.A+.Id
A rA[f]
f+ f+++
rA
As in the previous case we note that (5.6) is a morphism of R-maps. Thus functorality of the choice of diagonal factorisations establishes the requisite square’s commutativity, as it is precisely the left hand commutative square of (5.5).
3. Commutativity of
∆.A[f].A[f]+.Id
A[f] ∆.A[f].A[f]+.IdA[f].C[f+++]
Γ.A.A+.IdA Γ.A.A+.IdA.C
J(C[f],d[f])
f+++ f++++
J(C,d)
First note by functorality the commutative square of 2. is a morphism of L-maps from rA[f]
torA. By naturality of canonical fillers with respect to morphisms of clovenL-maps we have that
J(C, d)◦f+++is the canonical filler of the square:
∆.A[f] Γ.A.A+.IdC
∆.A[f].A[f]+.Id
A[f] Γ.A.A+.IdC df+
rA[f] πC
f+++
Similarly we have that the commutative square
∆.A[f].A[f]+.IdA[f].C[f] Γ.A.A+.IdA.C
∆.A[f].A[f]+.IdA[f] Γ.A.A+.IdA f++++
πC[f+++ ] πC
underlies a morphism of clovenR-maps. Using naturality of canonical fillers with respect to cloven
R-maps we have thatf++++◦J(C[f], d[f]) is the canonical filler of the square.
∆.A[f] Γ.A.A+.IdC
∆.A[f].A[f]+.Id
A[f] Γ.A.A+.IdC f++++d[f]
rA[f] πC
f+++
Observing thatdf+=f++++d[f] thus gives the required commutativity.