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So right now in my formal classes, I'm doing homology, and it looks to me like singular homology is just the homology of the complex:
.
However, it's not clear to me how to use maps to induce the boundary maps functorially using Yoneda. Is there any way to do it, or am I doing something that won't work? (note: F denotes the free abelian group on that set.)
The step you need is the Dold-Kan correspondence: what we get naturally from the boundary maps is a simplicial abelian group
You want to turn this into a complex by forming the Moore complex of this: the differential is the alterating sum of the maps induced by the boundary maps.
Freaking awesome. That is so cool! You've exceeded my expectations!
Thank you, I really appreciate it. This is exactly what I was looking for!
Maybe it is me who uses the terminology incorrectly. See the discussion at Moore complex -- related complexes.
I don't know. Somehow a word is missing in the standard literature! People like to say "normalized chains complex" when they mean the normalized version. There seems to be no good word for the non-normalized version apart from "the complex". Maybe: "that complex, you know" would be an option.
"that complex, you know"
:D
One could say, 'the Dold-Kan complex associated to a simplicial abelian group', but it is a bit long. I came to the Moore complex via the nonabelian version, so that is where I know it from. In any case, the homology of the Moore complex and "that complex" are isomorphic (naturally so I think), so it is a fine point.
By the way, David, you probably accidentally switched your text filter (happens to me every now and then, too). When posting comments, below the edit box choose "Markdown" and then your intended formatting will again appear.
Thanks - I sometimes turn it off so I can do quicker maths comments without using proper LaTeX.
Oh, i see. Never mind me, then. I suspected that you were like me and did this accidentally! ;.)
Yeah, I accidentally forgot to turn them back on!
Does anybody else find the notion of alternating signs distasteful? (Not to sound too picky), but is there perhaps some sort of "better" structure than imposing strange sign conditions?
I mean, in the case of, say, deRham cohomology, the signs come from the alternating form, but why should they come up in singular homology?
One way to think of it is:
the Dold-Kan correspondence allows us to regard everything related to chain complexes as just a tool and convenient repackaging of the true structure, which are abelian Kan complexes = abelian oo-groupoids.
From this perspective a chain complex has no further intrinsic meaning and is just a convenient way to encode an abelian oo-groupoids. If it takes alternating signs or whatnot to achieve this, then so be it.
But of course the signs are not all that arbitrary and pointless. For one, the alternating sum of face maps is precisely the sum of incoming face maps minus the sum of outgoing face maps. So what happens under Dold-Kan is also secretly a step from simplicial shapes to globular shapes. This is more manifest in the non-abelian version of Dold-Kan, which leads not to chain complexes but to crossed complexes. It is also manifest in the beginning of Ross Street's text on orientals.
Incoming face maps and outgoing face maps? I'm not sure that I'm familiar with the terminology, (or I could be missing the point!).
Here's my guess:
Take a 2-simplex: two of the faces (=edges if you draw it) are source and one is target if you take the simplex (=filler) as a globular shape. For higher simplices there should be a systematic way to say which faces are a component of the source and which are a component of the target. Orientals sounds a good way to do this.
Yes. Look at the pcitures at oriental. An oriental is an "oriented simplex", sort of: is is a simplex thought of as a globular n-category, where each k-morphism has a source and a target (k-1)-morphism.
You can check that the face maps that appear with a plus-sign in the alternating sum all correspond to source and those with a minus sign all to target, in any given situation.
The way I draw the 2-simplex, I get one vertex that is a source for two edges, one that is the source for one edge and target of another, and one that is the target for two edges.
The triangle where all edges are facing rightwards Edit: (Markdown broke my Ascii art)
Edit: Aren't simplices oriented by definition?
Is it worth reading over Goerss-Jardine for the answer to this, or is it not going to give the moral story?
This is not supposed to be a deep point that I am making here.
For n=2: notice that the two even faces can be composed and the 2-simplex may be regarded as going from the two even faces to the single odd face.
Similarly for the 3-simplex: the picture given at oriental shows a 3-morphism going from the two even faces to the two odd faces.
For the 4-simplex, look at that big picture at oriental. The big vertical arrow in the middle goes from the three even faces to the two odd faces.
For a full formalization of this you can look at Ross Street's The algebra of oriented simplices . But I don't think you want that fully formalized statement.
I couldn't see the picture for some reason! I had to open it in a different browser!
2 simplex = filled triangle, and it is the source and target of the putative 2-arrow that is that filler I was referring to, but Urs has given a better reference.
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