The model structure is not due to me but to Christian Sattler, and is explained in Section 2
(where I did some joint work is in the constructive argument that we have fibrant universes).
Cisinski defines in his book a model structure on presheaves where cofibrations
are monos. In order to define the model structure, one needs a “cylindre fonctoriel”
that we can define from a “segment” II (a presheaf with two global distinct elements 0 and 1)
Exemple 1.3.8. We further need a “donnee homotopique” Definition 1.3.14, given by a
set of monomorphism S. We only consider here the least case where S = empty set. From
this we can define a notion of anodyne map.
Given II and S, Cisinski explains then how to define a Quillen model structure Theorem 1.3.22
-the cofibrations are the monomorphism
-the anodyne maps are generated by open box inclusion, A x II \cup B x b -> B x II for A -> B
mono and b = 0,1
-the fibrant objects are the ones having the extension property w.r.t. anodyne maps
-the -naive- fibrations are the maps having the right lifting properties w.r.t. anodyne maps
(so a map X -> 1 is a naive fibration iff X if fibrant)
-the weak equivalence are the maps A -> B such that for any fibrant X, the map
X^B -> X^A is a homotopy equivalence (w.r.t. the choice of the interval II)
-the fibrations are the maps having the right lifting properties w.r.t. maps that are monos
and weak equivalence
W.r.t. the slides at
if we limit ourselves to the case where the base category is the Lawvere theory of
distributive lattices, or de Morgan algebra, or Boolean algebra and we take the fibrations
as defined page 13, we get the same notion as Cisinski -naive- fibrations.
It follows from Section 2 of Christian’s paper above that we also get a model structure
having exactly this notion of fibration (and where cofibrations are monos).
We then have a priori two model structures on these presheaves.
They have the same notion
of fibrant objects and cofibrations. It follows then from André Joyal’s result
(Riehl Categorical Model Theory, Theorem 15.3.1) that we actually the -same- model
It then follows that in these cases, the Cisinski model structure is -complete-
In the case of de Morgan algebra, or distributive lattice, we have a geometric realisation map.
However in the case of de Morgan algebra, we don’t have that this map is a Quillen equivalence
as shown by Christian: if we take L to be the quotient of II by the de Morgan reversal operation
then the geometric realisation of L is contractible, but it can be shown that the map L -> 1
is not an equivalence for Cisinski (= Christian in this case) model structure.
It is an open problem what happens for distributive lattices.
I am not a specialist of test categories but my understanding is that all these categories
(Lawvere theory of distributive lattices, de Morgan algebra, algebra with two constants 0,1)
are test categories, but it is not clear if the general notion of weak equivalence we get
from the theory of test category coincide with the one of Cisinski model structure
(and indeed it will not coincide for de Morgan algebra and Boolean algebra, and this
is open for distributive lattices),