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I'm confused between the definition of a simplex and a simplicial complex, as well as a face and a chain, and all of their various definitions of each.

(Definition 0): A concrete simplicial complex is the set of points $x\in \mathbb{R}^n$ such that $$x=\sum_{i=0}^n t_i a_i \quad\quad \sum_{i=0}^n t_i=1$$ and $\{a_0,\ldots, a_n\}$ is affine independent. This definition includes $\Delta^n$ as the favorite example.

(Definition 1):A singular $n$-simplex on $X$ is a continuous map $\sigma$: $$\sigma:\Delta^n\to X$$ where presumably $\Delta^n$ here is what it means in definition $1$.

(Definition 2): A simplicial complex $K$ in $\mathbb{R}^n$ is a collection of simplices in $\mathbb{R}^n$ such that

(i) Every face of a simplex in $K$ is in $K$.

(ii) The intersection of any two simplexes in $K$ is a face of each of them.

(Definition 3)We let $C_n(X)$ denote the free abelian group with basis the set of singular $n-$simplices in $X$. I've heard this called the simplicial complex. (Definition 3.5) I've also seen it where it's defined as $C_n(K)$ where $K$ is a simplicial complex..... and my understading was that it was the collection of faces (where it's clear that the face of a face is a face). I assume $3$ and $3.5$ are equivalent.

(Definition 4) An abstract simplicial complex is a collection $\mathcal{S}$ of finite non-empty sets, such that if $A$ is in $\mathcal{S}$, so is every nonempty subset of $A$.

(Definition 5) Let $K$ be a simplicial complex. A $p-$ chain is a function $c$ from the set of oriented $p-$simplices of $K$ to $\mathbb{Z}$ such that

(i) $c(\sigma)=-c(-\sigma)$ where $-\sigma$ is $\sigma$ with reversed orientation.

(ii) $c(\sigma)=0$ for all but finitely many oriented $p-$simplices $\sigma$.

Last thought outloud.... is $C_n(X)$ a group of functions or a group of abstract simplices from definition $0$ or $2$? I see definition $0$ and $2$ as roughly equivalent -- $(0)$ has a very concrete geometric picture whereas with $(2)$

So basically, I have all of these notions sort of floating around but not sure what is correct, maybe different authors have different definitions. But some clarity would be much appreciated.

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Basically the things you list look OK, with one big exception. A small point is that in (0) you should allow also simplices of lower dimension in $\mathbb R^n$. So you should talk about $k$-simplices in $\mathbb R^n$ for $0\leq k\leq n$ corresponding to points $a_0,\dots,a_k$ which are affinely independent. Doing this, the definition of a simplicial complex in (2) is fine. Then you just have to realize that if you look at the set $\mathcal S$ of all edges of all simplices contained in $K$, then any simplex in the complex corresponds to a subset $A\subset \mathcal S$ and the essential information about $K$ is contained in the list of subsets $A$. This leads to the relation between concrete and abstract simplicial complexes.

Now the big problem, which in my opinion causes all the confusion is that there is a second notion of "complex" basically derivied from "chain complex" (which or course originates from a construction for simplical complexes). The basic meaning of "complex" in this setting is "sequence of homomorphisms between abelian groups (or modules over a ring) such that the composition of two subsequent homomorphisms is zero". The homomorphisms are usuall called "differentials" or "boundary operators".

Now if you take a topological space $X$, the groups $C_n(X)$ and the boundary operators $d:C_n(X)\to C_{n-1}(X)$ form a complex in the latter sense, which is usually called the "singular complex" of $X$ and computes singular homology. Of course, for a simplicial complex $K$ you can do the same thing with the simplices of $K$ instead of all singular simplices (so you have finitely many generators overall instead of uncountably many generators in any dimension). The resulting homology is calle "simplical homology" (and is isomorphic to singular homology). Of course you could call the corresponding complex "the simplicial complex of the simplicial complex $K$", but the danger of confusion is getting very high here ...

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  • $\begingroup$ And now today I've come to the other definition... The "big problem". I thought it was quite some time away and some advanced topic. I will need more time to think about all this. But I appreciate your answer. $\endgroup$ – Clclstdnt Nov 20 '16 at 22:22

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