One of the common definitions of homology using the singular chains, i.e. maps from the simplex into your space. The free abelian group on these can be made into a chain complex and one can take the homology of this. The result is usually called singular homology.


However, one can use a smaller chain complex instead, by taking the quotient with the degenerate chains (those that are the image of a degeneracy map $\sigma$). This will give the same result in homology.


In some articles, I've seen authors replace the singular chains with the normalized singular chains, often claiming that this is "for technical reasons" (a example is Costello's article on the Gromov-Witten potential associated to a TCFT). What are the important technical differences between these two functors? In which situations is there a preferred one?



What are normalized singular chains good for?
Before going to linearity, note a technical advantage of normalized geometric realization is that it preserves products. That probably has linear consequences.


The simplest technical advantage of normalized chains is the [Dold-Kan theorem](http://ncatlab.org/nlab/show/Dold-Kan+correspondence) that the normalized chains give an equivalence of categories between simplicial abelian groups and chain complexes in positive degrees. But the tensor structures on the two categories (detailed in Tom's answer) [do not match](http://ncatlab.org/nlab/show/monoidal+Dold-Kan+correspondence) under this equivalence. The normalized functor is [both](http://golem.ph.utexas.edu/category/2009/11/doldkan_question.html#c029668) lax monoidal and oplax monoidal. I think the fat chain functor is still oplax monoidal. The technical advantage is that only the normalized functor is lax monoidal. My guess is that this is what Costello wants, in particular the consequence Tom mentions that a simplicial ring becomes a dga. Greg's parenthetical is another multiplicative property that I would expect to demonstrate the failure of the lax monoidal property, but I don't see that it does.


I have never seen an example where abnormal chains are *technically* advantageous; degenerate simplices tend only to get in the way of precise considerations. But if they don't get in the way, thinking about them may be a distraction, as Tom says.


