
\begin{rmk}In the references \cite{MV}, \cite{VSelecta} and \cite{Blander} (for $\mathbb{A}^1$-theory) and \cite{KSY2} (for the modulus-theory), the interval objects considered are always representable (either by $\mathbb{A}^1$ or by the modulus pair $(\P^1, 1)$). Here, we are pushing the ideas of \cite{KSY2} further to get a theory that works more generally for interval objects in categories of presheaves that are not necessarily representable.
    \end{rmk}
\subsubsection{}\label{subsubsec:canonical-maps-btw-intervals}Let $\mathbb{A}^1$ denote the representable simplicial presheaf $h_{(\mathbb{A}^1;\emptyset, \emptyset)} = h_{u(\mathbb{A}^1, \emptyset)}$. It is clearly an interval object in ${\mathbf{\ol{M}}}\mathcal{M}(k)$ for the cartesian product as well as for Day convolution product, since by \ref{subsubsec:tensor-cart-empty-modulus} we have $\mathbb{A}^1\otimes \mathbb{A}^1 = \mathbb{A}^1\times \mathbb{A}^1$.  
From the admissible morphism $j\colon (\mathbb{A}^1, \emptyset)\hookrightarrow (\P^1, \infty)$ in $\mathbf{Sm}_{\rm log}(k)$ we get maps of motivic spaces \[\mathbb{A}^1\to {\ol{\square}}\to \mathcal{I} \text{ in }{\mathbf{\ol{M}}}\mathcal{M}(k),\quad \mathbb{A}^1\to {\ol{\square}}\to v^*({\ol{\square}}) \text{ in }\cM_{\rm log}(k) \]
that we will use to compare the different interval structures on ${\mathbf{\ol{M}}}\mathcal{M}(k)$, on $\cM_{\rm log}(k)$ and on $\mathcal{M}(k)$.

\subsubsection{}We start by comparing $\mathbb{A}^1$ and $v^*({\ol{\square}})$ with the standard interval $\mathbb{A}^1$ in the category of motivic spaces $\mathcal{M}(k)$. Recall that the adjoint pair $\lambda\colon \mathbf{Sm}(k)\leftrightarrows \mathbf{Sm}_{\rm log}(k)\colon \omega$ gives rise to a string of four adjoint functors
\[(\lambda_!, \lambda^* = \omega_!,\lambda_* = \omega^*, \omega_*), \quad \omega_!=\lambda^*\colon \cM_{\rm log}(k)\leftrightarrows \mathcal{M}(k)\colon \lambda_*\]
(i.e. the functor $\omega_!$ has in turn a left adjoint). The functors $\lambda^*$ and $\omega^*$ clearly commute with products. Since $\lambda$ commutes with products and $\lambda_!$ commutes with Yoneda, the same argument used in the proof of Proposition \ref{prop:I-is-interval} (or even Proposition \ref{prop:ImKelly} directly) shows that $\lambda_!$ is monoidal with respect to the cartesian product. 
\begin{lem}\label{lem:comparison-A1-box-M}There are canonical isomorphisms $\lambda^*(\mathbb{A}^1) \cong \lambda^* v^*({\ol{\square}}) \cong \mathbb{A}^1$ as interval objects of $\mathcal{M}(k)$.
    \begin{proof}Since $\lambda^* = \omega_!$, it's clear that $\lambda^*(\mathbb{A}^1) = \omega_!(h_{(\mathbb{A}^1, \emptyset)}) = h_{\omega(\mathbb{A}^1, \emptyset)} = h_{\mathbb{A}^1} = \mathbb{A}^1$. For the second statement, it's enough to check that $\lambda^*v^*({\ol{\square}}) \cong\mathbb{A}^1$ as presheaves on $\mathbf{Sm}(k)$, and we just have to play with adjunctions. For $X\in \mathbf{Sm}(k)$, we have
        \begin{align*}\hom_{\mathbf{Psh}(\mathbf{Sm}(k))}( h_X, \lambda^*v^*({\ol{\square}})) &= \hom_{\mathbf{Psh}(\mathbf{Sm}_{\rm log}(k))}(\lambda_!(h_X), v^*({\ol{\square}})) \\ &= \hom_{\mathbf{Psh}(\mathbf{BSm}_{\rm log}(k))}(v_!(h_{(X, \emptyset)}), {\ol{\square}}) \\ &=\hom_{{\mathbf{BSm}_{\rm log}(k)}}((X, \emptyset), (\P^1, \infty)) \\ &= \hom_{\mathbf{Sm}(k)}(X,  \mathbb{A}^1).
        \end{align*}
The fact that the isomorphisms are compatible with the interval structure is a tautology from the definitions.
        \end{proof}
    \end{lem}
\begin{rmk}Note that the interval $\mathcal{I}$ of ${\mathbf{\ol{M}}}\mathcal{M}(k)$ is obtained from $v^*({\ol{\square}})$ by applying the left adjoint $u_!$ of the restriction functor $u^*$. The exact same construction, using $\omega_!=\lambda^*$ instead of $u_!$ produces, in view of Lemma \ref{lem:comparison-A1-box-M}, nothing but the usual interval $\mathbb{A}^1$ in the Morel-Voevodsky category of motivic spaces.
    \end{rmk}
\subsubsection{}Before moving forward to give more refined comparisons, we ask the following question. Let $F\colon \mathbf{Sm}_{\rm log}(k)\to \mathbf{Sm}(k)$ be the forgetful functor $(X,\partial X)\to X$. Then $F$ defines the usual set of adjoint functors between the categories of spaces $\cM_{\rm log}(k)$ and $\mathcal{M}(k)$, namely $(F_!, F^*, F_*)$. The general principle illustrated above allows us to construct the object $J\in \mathcal{M}$ as $J=F_!(v^*({\ol{\square}}))$. Since $F$ is strict monoidal for the cartesian product, the argument of Proposition \ref{prop:I-is-interval} goes through to show that $J$ is in fact an interval object in $\mathcal{M}(k)$ for the usual product of simplicial presheaves.

The canonical adjunction map $\eta\colon {\ol{\square}}\to v^*({\ol{\square}})$ gives then a map $\P^1\to J$ in $\mathcal{M}$. By construction, this map cannot be an isomorphism (since $\P^1$ does not have any interval structure on $\mathcal{M}(k)$), and is also easy to see that $J$ is not isomorphic to $\mathbb{A}^1$. One could therefore try to develop a machinery for the localization of $\mathcal{M}(k)$ to $J$, in the same spirit of what we will do for ${\mathbf{\ol{M}}}\mathcal{M}(k)$ with $\mathcal{I}$. We don't know, at the moment, what would be the outcome of such a construction. 

\section{Motivic homotopy categories with modulus}
\subsection{Model structures on simplicial presheaves}
The category of simplicial sets $\mathcal{S}$ carries a well-known cofibrantly generated model structure (the Quillen model structure). 
The category of simplicial presheaves $s\mathbf{Psh}(T)$ on a small Grothendieck site $T$ carries two natural model structures, the \textit{injective} and the \textit{projective} model structure presenting the same homotopy category, i.e. having the same class of weak equivalences:
\begin{df}A map $f\colon A\to B$ in $s\mathbf{Psh}(T)$ in $s\mathbf{Psh}(T)$ is called an \textit{objectwise} (or \textit{levelwise} or \textit{sectionwise}) \textit{simplicial weak equivalence} if $f(X)\colon A(X)\to B(X)$ is a weak equivalence of simplicial sets for each $X\in T$. A map $f\colon A\to B$ is called an \textit{objectwise Kan fibration} if $f(X)\colon A(X)\to B(X)$ is a Kan fibration of simplicial sets for each $X\in T$.
    \end{df}
 The injective model structure on $s\mathbf{Psh}(T)$ was introduced by Heller in \cite{Heller}, in which the cofibrations are the monomorphisms, and the fibrations are the maps having the right lifting property with respect to the trivial cofibrations. It is a proper, simplicial cellular model structure.
We will refer to the fibrations for the injective model structure as the \textit{(simplicial) injective fibrations}. The category of simplicial presheaves with the injective model stucture will be denoted, following the usual convention, $s\mathbf{Psh}(T)_{\rm inj}$. Note that every object is cofibrant for the injective structure