在代数拓扑书上看到：
吓惨了！怎么就一个是和一个是积呢？这个书是错不了的。the cohomology group H^1(X;G) is a direct product of copies of the group G, ... This can be compared with the homology group H_1(X;G) which consists of a direct sum of copies of G....
网上找到这个通俗版的解释：
In general, the difference between a product structure and a coproduct (i.e. sum) structure is the direction it relates to its components.When you have a product structure, you have 'projections' A×B->A and A×B->B, as well as the fact that any pair of maps C->A and C->B can be lifted to a map C-> A×B in exactly one way.
When you have a sum structure, you have 'inclusions' A->A+B and B->A+B, as well as the fact that any pair of maps A->C and B->C can be pushed to a map A+B->C in exactly one way.
The thing that makes linear algebra really cool is that A×B=A+B for vector spaces and other similar structures, as adriank said. (Though this doesn't remain true when you compare an infinite product of structures to an infinite sum of structures)
(link)
最终wiki上面有详细的解释。
http://en.wikipedia.org/wiki/Direct_product
http://en.wikipedia.org/wiki/Product_(category_theory)
http://en.wikipedia.org/wiki/Coproduct
我想我大概有个好习题：一般拓扑书上讲的那个无穷乘积拓扑是直积，不是直和。
同调上同调这个我就想当然了。呵呵
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