Definition of a Δ-set?

Question

According to Wikipedia,

Formally, a Δ-set is a sequence of sets $\{S_{n}\}_{{n=0}}^{{\infty}}$ together with maps

${\displaystyle d_{i}\colon S_{n+1}\rightarrow S_{n}}$

with ${\displaystyle i=0,1,\ldots ,n+1}$ for $n\geq 1$ that satisfy

$d_{i}\circ d_{j}=d_{{j-1}}\circ d_{i}$

whenever $i<j$.

There are multiple things that are unclear to me with this definition.

1. The wording implies that the sequence of sets is together with maps $d_{i}$, but this doesn't make sense, as $n$ has a range up to $∞$ whereas $d_{i}$ is defined as a finite family in terms of $n$. Am I correct to understand that it's the sets that each have an associated family of maps with them? (The diagram seems to support this interpretation: https://upload.wikimedia.org/wikipedia/commons/9/94/Delta_maps.svg ; ignore the lower part.)

2. The map composition doesn't make sense to me. The domain is always larger than the codomain, so clearly the maps $d_{i}$ and $d_{j}$ can't be composed unless they belong to different families, of $S_{n}$ and $S_{n+1}$, respectively. Is this implicitly assumed?

3. But if this is the case, it does't make sense to compose $d_{i}$ as both the inner and outer function, so surely something else is meant with the composition? Or is $d_{j-1}$, again, implicitly assumed to belong to a different family of $S_{n-1}$?

Answer 1

The notation used there is indeed rather confusing if you have not seen these objects before. Really you should say that there are maps $d_i^n:S_{n+1}\to S_n$ for each $n\geq 0$ and $i=0,1,\dots,n+1$. Then the equation they are supposed to satisfy is $d_i^{n-1}\circ d_j^n=d_{j-1}^{n-1}d_i^n$. (Note also that Wikipedia's definition erroneously says these maps should exist only for $n\geq 1$, rather than $n\geq 0$. I imagine this was the result of some indexing mix-up where at some point the maps were going from $S_n$ to $S_{n-1}$ instead of from $S_{n+1}$ to $S_n$.)

In practice, when working with these maps, people often omit the superscripts since they get cumbersome and there is typically only one choice for them that makes sense in context. You shouldn't do this when making the initial definition of a $\Delta$-set, though!

Answer 2

Remarks $(2)$ and $(3)$ are both due to the suppressed indices or suppressed grading. Really, this is a family of set $S_n,n\in\Bbb N_0$, with functions $d_{i,n}:S_n\to S_{n-1}$ for $n\ge1$ and $0\le i\le n$. The comment in $(1)$ about "$n$ goes to infinity" doesn't make sense to me but hopefully your doubt will be cleared once you realise the $d_i$ are actually different functions for different $n$; it is implicit.

The composition conditions do not feature the same $d_i$ on both sides! The indices shift, which resolves any issues. $$d_{i,n-1}\circ d_{j,n}=d_{j-1,n-1}\circ d_{i,n},\,0\le i<j\le n$$

Answer 3

Basically they're simplicial complexes but you're allowed to have loops and multiple edges (as well as higher dimensional analogues of these). So think about the definition in those terms. $S_n$ is the set of $n$-dimensional "simplices" and $d_0, \dots, d_n : S_n \to S_{n - 1}$ are the maps taking an $n$-simplex to faces numbered $0, \dots, n$.

So in words:

• You have a collection of simplices of various dimensions. Each of those simplices has faces which are numbered (face 0, face 1, ... , face n). E.g. a triangle would have a 0th, 1st, 2nd edge.

• The concept of say the "0th face" defines a distinct map $d_0 : S_n \to S_{n-1}$ for each $n$. In terms of putting all the data in, we could write this as $d_0^n$ to indicate that its domain is $S_n$.

• The identity $d_i \circ d_j = d_{j - 1} \circ d_i$ says:

the $i$th face of the $j$th face equals the $(j-1)$st face of the $i$th face.

And just like how there are different $d_0$'s for each $n$, the concept of "face" also changes depending on $n$ like an edge is a face of a triangle and a point is a face of an edge.