Let $J$ be an interval of integers viewed as a linearly ordered set, and let $I \subseteq \mathbf{N}(J)$ be the subsimplicial set given by the union of the elementary edges $(x, x+1)$.


The inclusion $I \to \mathbf{N}(J)$ is a categorical equivalence, and so for a quasi-category $\mathcal{C}$, functors $\mathbf{N}(J) \to \mathcal{C}$ can be expressed up to homotopy by maps $I \to \mathcal{C}$ of simplicial sets.


Now, consider the two-dimensional case. Products preserve categorical equivalences, so functors $\mathbf{N}(J \times J) \to \mathcal{C}$ can be expressed, up to homotopy, by maps of simplicial sets $I \times I \to \mathcal{C}$ — or equivalently, it is enough to specify just the images of the individual squares
$$ \begin{matrix} (x,y) &\to& (x+1, y)
\\ \downarrow & & \downarrow
\\ (x,y+1) &\to& (x+1, y+1)
\end{matrix} $$


My question is about subdiagrams of this; what if I'm interested in some sub-poset $D \subseteq J \times J$ made out of a collection of these elementary squares? Is it always enough to specify a functor $\mathbf{N}(D) \to \mathcal{C}$ simply by specifying the images of the squares?


If not in general, what about the special case of the indexing category $J^{\Delta[1]}$ used in the definition of $J$-complexes in pointed $\infty$-categories?



Specifying complexes in quasicategories via squares
The specific example of presenting $J^{\Delta[1]}$ can be resolved by the trick of observing it is a retract of $J \times J$.


Let $K$ be the subsimplicial set of $I \times I$ consisting of:


* The vertices $(x,y)$ with $x \leq y$
* For each $x \leq y < \max(I)$, the elementary square with top-left vertex $(x,y)$
* For each $x < \max(I)$, the triangle spanned by $(x,x) \to (x,x+1) \to (x+1, x+1)$


The inclusion $J^{\Delta[1]} \to J \times J$ is split by the map $(m,n) \mapsto (m, \max(m, n))$, and after taking nerves, this retraction diagram restricts to a retraction $K \to I \times I \to K$.


Since $I \times I \to \mathbf{N}(J \times J)$ is a categorical equivalence, the restriction $K \to \mathbf{N}(J^{\Delta[1]})$ is also a categorical equivalence, by the general property of model categories that retracts preserve the weak equivalences.


