Walk-through to Morel-Voevodsky A¹-homotopy theory, page 48-50

We look at the model structure Voevodsky and Morel use in their 1999 IHES paper and discuss 1.2, 1.3, 1.4, 1.5, 1.6, 1.8, 1.9, 1.10. There is nothing difficult or particularly interesting, but you might want to look up some specific issue or reference.
I wrote another posting that explains what an enriched model category, enriched over a monoidal model category is; we turn to simplicial model categories in this post. There, I also explain the notion of monoidal and enriched model categories beside some notions of simplicial sets and topoi, the most important being for now:

a simplicial model category is just an enriched model category which is enriched over the monoidal model category of simplicial sets.

but details are also to be found below.

The simplicial model structure on simplicial sheaves on a topos

In Definition 1.2, for every small site $T$ , a model structure on $\Delta^{op}Shv(T)$ is defined:

The weak equivalences $W_s$ are the stalkwise (pointwise) weak equivalences
The cofibrations $C$ are the monomorphisms
The fibrations $F_s$ are defined via the right lifting property with respect to acyclic cofibrations

Remark 1.3 is a technical subtlety. If you happen to have a conservative set of points $P$ of a topos $T$ , then weak equivalence of a morphism $f : X \rightarrow Y$ of sheaves on $T$ can be tested pointwise: $f \in W_s \Leftrightarrow \forall x^\ast \in P : x^\ast(f) \in W$ , where $W$ denotes the weak equivalences in the standard model structure of simplicial sets. A conservative set of points $P$ is just a set of points that is a conservative family of functors, which is by definition, that the product functor $\prod_{x \in P} x$ is a conservative functor.
A functor $F$ is conservative if it reflects isomorphisms. That means, $F(f)$ isomorphism implies $f$ isomorphism for each morphism $f$ .
This technical lemma is used later in the text, but the homotopy sheaves are not, so I guess you can forget the proof details when reading the text for the first time.

See also: conservative functor in nLab
Theorem 1.4 (the structure defined by $(W_s,C,F_s)$ is a model category structure) cites the result of Corollary 2.7 in Jardine: Simplicial Presheaves, in no. 47 J.Pure Applied Math, 1987 which is originally due to Joyal. Since the article is behind a paywall, I'll give you a rough idea:

(MC1), (MC2) and (MC3) are deduced from the model structure on simplicial sets.
(MC4) relies on the fact that the morphism from a presheaf to its associated sheaf is a weak equivalence and then applying the axiom for $\Delta^{op}Preshv(T)$ with the global fibration and topological weak equivalence model structure. (MC4) for $\Delta^{op}Preshv(T)$ is proved with a trick that uses (MC5).
(MC5) is essentially a small object argument.

The corresponding homotopy category of $(W_s,C,F_s)$ on $\Delta^{op}Shv(T)$ is written $\mathcal{H}_s(T)$ .

Proper model categories

Remark 1.5 states that the model structure is a proper one. The proof is available in Jardine, J.F.: Stable homotopy theory of simplicial presheaves, in no. 39 Can. Math. J, 1987 which is available for free here.

A simplicial model category is proper if

(P1) the pullback $j^\ast(g)$ of a weak equivalence $g$ along a fibration $j$ is always a weak equivalence,
(P2) the pushout $i_\ast(f)$ of a weak equivalence $f$ along a cofibration $i$ is always a weak equivalence.

(P1) is proved for simplicial sets via fibrant replacement, such that one has a cartesian diagram up to weak equivalence, and then application of K. Brown's coglueing lemma, which is Lemma 1 on page 428 of Brown, K.: Abstract Homotopy Theory and Generalized Sheaf Cohomology, in Vol. 186 Transactions of the American Mathematical Society, 1973 which you can download from the nLab for free.
(P2) is proved for simplicial sets in a dual fashion, using the fact that simplicial sets are always cofibrant and a dual of Brown's coglueing lemma.

For simplicial presheaves on a topos, the proofs are similar. For (P1), fibrant replacement yields a cartesian diagram (up to weak equivalence) in which all objects are locally fibrant simplicial presheaves (which form a category of fibrant objects) and the coglueing argument can be applied. For simplicial sheaves, (P1) and (P2) follow since the associated sheaf morphism is a weak equivalence.

It should be mentioned that (P1) is also called right proper and similarly (P1) left proper.

Functorial fibrant replacements (1.6)

(MC5) demands in particular, that every morphism is functorially factorizable into a fibration after an acylic cofibration.
A resolution on a site $T$ (which carries a model structure) is defined to be a functor $Ex : \Delta^{op}Shv(T) \rightarrow \Delta^{op}Shv(T)$ and a transformation $\theta : Id \rightarrow Ex$ such that for every simplicial sheaf $X \in \Delta^{op}Shv(T)$ , the object $Ex(X)$ is fibrant and $\theta_X : X \rightarrow Ex(X)$ is an acyclic cofibration.
Indeed, if $f : X \to \ast$ is a morphism, we can factorize it into an acyclic cofibration followed by a fibration. Rename the acyclic cofibration $\theta_X$ and the object $\theta_X(X) =: Ex(X)$ , then $Ex(X) \rightarrow \ast$ is a fibration, thus $Ex(X)$ fibrant. Voilà - since (MC5) demands this to be functorial, the functor/transformation conditions for a resolution are fulfilled.
It should be clear that this works the same way for cofibrant replacements, although we won't need this here, since in the simplicial model structure we're looking at on $\Delta^{op}Shv(T)$ , all objects are cofibrant.

Simplicial model categories

For every two objects $X,\ Y \in \Delta^{op}Shv(T)$ , we defined $! S(X,Y) := Hom(X \times \Delta^\bullet,Y) \in \Delta^{op}Set.$
$S(X,Y)$ is a simplicial set because $\Delta^\bullet$ is a cosimplicial object. If you take an object $U \in T$ as constant simplicial sheaf in degree 0, you can look at $S(U,X)$ , which is just the simplicial set of sections $X(U)$ for the simplicial sheaf $X$ . Now we have to see that this enrichment is compatible with the model structure. This is done in Remark 1.9. resp. Lemma 1.8. The proof indication for Lemma 1.8. is to prove 1) via points of $T$ . This is easy if you already know that the standard model structure on simplicial sets is a simplicial model structure (the model category of simplicial sets enriched over the monoidal model category of simplicial sets), which is not too hard to prove.

If you already know about the "subtleties" in the definition of simplicial model categories (maybe from my article about simplicial model categories), skip the next two paragraphs.

A category $\mathcal{C}$ is a simplicial model category if it is a model category that is enriched over simplicial sets, that satisfies the additional axioms (Quillen):

(SM0): for all $X \in \mathcal{C}$ and all finite simplicial sets $K$ , $X \otimes K$ and $X^K$ exist.
(SM7): If $i: A \rightarrow B$ is a cofibration and $p:X \rightarrow Y$ a fibration, then $! \underline{Hom}(B,X) \xrightarrow{(i^\ast,p_\ast)} S(A,X) \coprod_{S(A,Y)} S(B,Y)$ is a fibration of simplicial sets, which is trivial if either $i$ or $p$ is trivial. (The S denotes the simplicial mapping object of $\mathcal{C}$ ).

(SM0) is also phrased " $X$ is powered and copowered" and sometimes already included in the definition of an enriched model category (like I did in my article about simplicial model categories). (SM7) is also phrased "the copower functor is a left Quillen bifunctor" and sometimes already included in the definition of an enriched model category (like I did, again). So, if you take the "modern" definition of a model category enriched over a monoidal model category, those axioms are already included (I put them in here just because they will show up in the literature and also because you might not have read my article about the definition of simplicial model categories).

Lemma 1.10, different notions of equivalence are the same

For $X,\ Y \in \Delta^{op}Shv(T)$ fibrant and $f:X\rightarrow Y$ a morphism, these three statements are equivalent:

$f$ is a simplicial homotopy equivalence,
$f$ is a weak equivalence,
$\forall U \in T : S(U,f)$ is a weak equivalence.

The proof indication is mostly a list of references, so let's have a more detailed look, which will then finish this posting.

(2)=>(1)
factorise the weak equivalence $f$ into a cofibration $i : X \rightarrow X'$ followed by an acyclic fibration $p : X' \rightarrow Y$ . Then $i$ is a weak equivalence again (by 2-out-of-3). By an argument in Quillen's Homotopical Algebra (Corollary 2.5), obtain a retraction $r$ of $i$ by the lift in the diagram
and then get a simplicial homotopy from $ir$ to $id_{X'}$ by the lift in the diagram
and now $r$ is a simplicial homotopy inverse of $i$ . To actually obtain a simplicial homotopy inverse of $f$ , we're going to build a simplicial homotopy inverse of $p$ . For this, observe that all objects are cofibrant (since cofibrations are by definition just monomorphisms), and that the dual statement to what we just proved is that a trivial fibration between cofibrant objects is a simplicial homotopy equivalence.
What is $I$ ? What is $X^I$ ? you might ask. The object $I$ is just the simplicial set $\Delta^1$ , whose geometric realisation in $\mathbb{R}$ looks like the interval $[0,1]$ , hence the name (and I used this notation here because it's the same as in Quillen's book). The object $X^I$ is the internal mapping object $\underline{Hom}(\Delta^1,X)$ . If this remains unclear, you might want to read some introduction to enriched category theory.
(1)=>(3)
We will not try to construct a weak homotopy equivalence but a homotopy equivalence:
Using the definition of $Y(U)=S(U,Y)$ for $U \in T$ and $Y \in \Delta^{op}Shv(T)$ , you'll see the canonical isomorphism $X^{\Delta^1}(U) \xrightarrow{\simeq} X(U)^{\Delta^1}$ . Now take a simplicial homotopy inverse $g$ to the map $f$ and choose a simplicial homotopy $h_X : X \rightarrow X^{\Delta^1}$ between $id_X$ and $gf$ . This yields a map $S(U,h_X) : X(U) \rightarrow X^{\Delta^1}(U)$ which, composed with the canonical isomorphism above, is the homotopy between $S(U,g)\circ S(U,f)$ and $id_{X(U)}$ we're looking for. The other composition $fg$ is handled the same way.
(3)=>(2)
From SGA4 6.8.2 we learn that every point $x^\ast$ of $T$ has an associated functor $Vois_T(x) \rightarrow T$ , where $Vois_T(x)$ is the category of neighbourhoods (French: voisinages) of $x^\ast$ . A neighbourhood is a couple $(U,u)$ where $U\in T$ and $u \in x^{\ast}U$ . The cofiltrant category of neighbourhoods of $x^\ast$ admits a small cofinite full subcategory, so by abstract nonsense the functor $Vois_T(x) \rightarrow T$ is a pro-object in $T$ . A pro-object is, by definition, just a functor from a small cofiltered category to $T$ (think of it as a diagram to form a projective limit, hence the name). Let's write the pro-object $\{U_\alpha\}$ , hiding the small cofinal full subcategory of $Vois_T(x)$ in the indices.
Now for a point $x^\ast$ , $x^\ast(f)$ is a filtering colimit (=projective limit) of all $S(U_\alpha, f)$ , thus a filtering colimit of weak equivalences. We conclude that $x^\ast$ is itself a weak equivalence. Since this holds for every point, $f$ is a weak equivalence.