I noticed something interesting, which may already be known, which is about 2-Segal spaces (or decomposition spaces) and presheaves on the category $\Delta S$. You'll need to already know something about simplicial sets before starting.

2-Segal spaces

The 2-Segal condition, introduced by Dyckerhoff & Kapranov, is a generalization of the usual Segal condition on a simplicial object $X$. The Segal condition says that the following square is a pullback:

Here I'm writing $d_\bot$ for the bottom face map $d_0$ and $d_\top$ for the top face map $d_{n+1}$ or $d_n$ This is perhaps not the usual way of presenting the Segal condition. More typically we might ask that the spine map gives an isomorphism \[ X_n \simeq X_1 \times_{X_0} \times \cdots \times_{X_0} X_1.\] But no matter, these turn out to be the same. For simplicial sets, the nerve theorem says that simplicial sets satisfying the Segal condition are precisely those that are isomorphic to nerves of categories.

What about the 2-Segal condition? Well I'm going to split it into two parts, the upper 2-Segal condition as on the left below, and the lower 2-Segal condition as on the right below. In both cases we are asking that the indicated commutative square is a pullback for all $0 < i < n$.

It turns out that if $X$ satisfies the Segal condition, then it also satisfies both the upper and lower 2-Segal conditions. Usually when people talk about 2-Segal spaces they mean that both hold. So we can think about the 2-Segal simplicial sets as a kind of generalization of categories. Having both upper and lower 2-Segal conditions makes it so that the span \[ X_1 \times X_1 \xleftarrow{d_2 \times d_0} X_2 \xrightarrow{d_1} X_1 \] endows $X_1$ with an associative multivalued multiplication.

There are a couple recent introductions to 2-Segal conditions, one by Walker Stern and one by me.

ΔS

The other player is the category $\Delta S$ which was introduced by Fiedorowicz and Loday. This is an enlargment of $\Delta$ which has the same objects but more morphisms, just like Connes' cyclic category $\Lambda$ does (we in fact have $\Delta \subset \Lambda \subset \Delta S$). The essence of this category is that the automorphism group of $[n]$ is enlarged. In $\Delta$, the automorphism group is the trivial group, but in $\Delta S$ it is the symmetric group $S_{n+1}$ (in Connes' cyclic category it is $C_{n+1}$). In particular, if $X$ is a presheaf on $\Delta S$ (and so in particular a presheaf on $\Delta$), then $X_n$ has an action by $S_{n+1}$.

The morphisms of this category are not just maps of sets $[n] \to [m]$ (which also has $S_{n+1}$ as the automorphism group). Rather, they're such maps together with a total order on fibers (see Pirashvili and Richter).

The category $\Delta S$ has a presentation by generators and relations much like $\Delta$, but throwing in additional generators $\theta^k$ for $(k,k+1) \in S_{n+1}$. I'll just give the relationship with $\delta^i$, since that's the one I need: \[ \theta^k \delta^i = \begin{cases} \delta^i \theta^k & k < i-1 \\ \delta^{i-1} & k = i-1 \\ \delta^{i+1} & k = i \\ \delta^i \theta^{k-1} & k > i. \end{cases} \] (This, along with the other relations, appears as exercise E.6.1.7 in Loday's book Cyclic Homology and is attributed to Clauwens.)

2-Segal ΔS presheaves are Segal

We now come to the punchline. Suppose we've got a presheaf $X \colon \Delta S^{\text{op}} \to C$, and assume that its restriction to $\Delta$ satisfies the upper or lower 2-Segal condition. Then it also satisfies the Segal condition. Why is this the case? Well, what it boils down to is that in $\Delta S$ the following two squares are isomorphic.

The isomorphism is given by the map (one map for each corner) \[ \begin{array}{c|c} id_{[n-1]} & \theta^{n-1} \\ \hline id_{[n]} & \theta^n \end{array} \] which you can check using the identity above. (It doesn't matter which direction you point this, it will still work out.) Now apply $X$. The isomorphism tells us that almost every square under consideration for the Segal condition is isomorphic to the $i=n-1$ upper 2-Segal square. Since we assumed the latter are all pullbacks, so are the former.

This doesn't quite do it: when $n+1$ is $2$, there is no $i = n - 1$ case of the upper 2-Segal condition, since we required $0 < i < n$. This can be handled via a retract argument from the $n+1=3$ Segal square.

Why did I think about this?

I'm currently working on a big project with Justin Lynd where we are studying relationships between partial groups and higher Segal spaces. There is some hierarchy of upper and lower $d$-Segal objects, based on higher-dimensional geometric shapes, with Segal and 2-Segal objects sitting at the bottom. Whatever partial groups are, it's very convenient to cast them as certain symmetric simplicial sets. By this, I don't mean presheaves on $\Delta S$, but rather the simpler version where your morphisms are just ordinary functions $[n] \to [m]$. We showed the following:

Theorem. Let $k\geq 1$. For symmetric simplicial objects, the lower $(2k-1)$-Segal, lower $2k$-Segal, upper $2k$-Segal, and upper $(2k+1)$-Segal conditions are all equivalent.

The ordinary Segal condition is called "lower 1-Segal" in this hierarchy, so the $k=1$ case of the theorem is closely related to our observation in this post. In fact, the $\Delta S$ observation is a bit stronger, since every symmetric simplicial set is automatically a $\Delta S$ presheaf, but not vice-versa.

We'll post our paper soon, but for now you can look at the extended abstract if you're curious about what we're up to.

Outlook

One might wonder about the above theorem for $\Delta S$-presheaves instead of symmetric sets. I think it's true! But the proof is more complicated, whereas the symmetric sets proof is very intuitive.

If you know about crossed simplicial groups, you might ask about whether the 2-Segal conditions also collapse to the Segal condition for other categories built from crossed simplicial groups. Well, for many of them this isn't true, and 2-Segal really is more general than Segal. Dyckerhoff and Kapranov have given some really cool examples for the cyclic case and for more general planar crossed simplicial groups.