In any presheaf topos, there exists an object called *Lawvere's segment*, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h\_a: x\in Ob([A^{op},Set])\}$. 


That is, $L$ assigns to each object $a\in Ob(A)$ the set of monomorphisms with target $h\_a(\cdot):=Hom(\cdot,a)$  

in $[A^{op},Set]$. Since all objects in $[A^{op},Set]$ can be given as colimits of the representable functors $h\_a$ for $a\in A$, we can actually give a characterization of the functor represented by $L$ as the "sub-object assigning functor". 


Lawvere's segment is useful, because it naturally has the structure of a cylinder functor (given by taking the cartesian product with a presheaf $X$). In fact, it is a theorem of Cisinski (very closely related to the specialization Jeff Smith's theorem to presheaf toposes) that every accessible localizer on a presheaf category admits the structure of a closed model category generated by the homotopical data (donnée homotopique is the term used by Cisinski, so this is my rough translation) of some *set* (as opposed to proper class) of arrows $S\subset Arr([A^{op},Set])$, some cellular model $\mathcal{M}$, and the canonical cylinder given by Lawvere's segment. 


Then my question: How can we describe Lawvere's segment in $Set\_\Delta$ explicitly, that is to say, geometrically? 


Edit: A cellular model M on a presheaf topos is a *set* of monomorphisms $M$ such that $llp(rlp(M))$ is the class of all monomorphisms. Note that $llp$ and $rlp$ give the class of arrows with the left lifting property (resp. right lifting property) to the class of arrows given in the argument. 



An explicit description of Lawvere's segment in the category of simplicial sets
I've never heard this called 'Lawvere's segment' before, but your $L$ is the subobject classifier in the presheaf topos $[A^{\mathrm{op}},\mathrm{Set}]$. In presheaf toposes generally, the subobject classifier $\Omega$ is the presheaf that sends an object $a$ to the set of all sieves on $a$, i.e. the set of subobjects of $\hom(-,a)$. This means in this case that simplicial subsets $S' \subset S$ are in bijection with simplicial maps $S \to L$.


Off the top of my head, I can't be exactly sure what $L$ looks like, but I'd guess it's the constant simplicial set with $L\_n = 2$ (the set of truth values) and all maps identities. Then the characteristic map $\chi\_{S'} \colon S \to L$ will be the usual $s \mapsto 1$ if $s \in S'$ and $0$ otherwise, while the simplicial subset corresponding to some $\phi \colon S \to L$ will be given by the fibres $\phi\_n^{-1}(1)$ in each degree. $S'$'s being a *simplicial* subset should correspond to $\phi$'s being a simplicial map.


I could easily be wrong about that last paragraph, though. There might be more about this in Mac Lane--Moerdijk.


Edit: I was wrong -- see the comments below.


