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I have to be sincere here and say that I still don't get the real difference between a product and a sum.

if I have the trivial order on a set of objects in a category. then the direct limit is a sum and the projective limit is a product.. I really don't see the difference between those two notions: they seem equivalent to me: in a vector space the direct sum of 1 dimensional spaces spanned by the basis vectors could very well be presented by a product.

I hope I was clear in describing my confusion.. I would like to know if even in the trivial case the two notions of direct and inverse limit are not same.

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As the tag description for algebra says, that is a tag for symbolic manipulation of expressions. Abstract-algebra fits better here. – Asaf Karagila May 12 '11 at 9:07
up vote 4 down vote accepted

Think of a poset as a category where $a \le b$ if and only if there is a single arrow $a \to b$, and otherwise there are no arrows. Verify that coproduct means maximum and product means minimum. More generally, colimit (in particular, direct limit) means supremum and limit (in particular, inverse limit) means infimum.

Your confusion might stem from the fact that for a finite collection of vector spaces the product and coproduct agree; this is because the category of vector spaces has finite biproducts, as does, for example, the category of abelian groups. However, this is false for infinite collections: the coproduct of countably many copies of a $1$-dimensional vector space has countable dimension, but the product of countably many copies of a $1$-dimensional vector space has uncountable dimension.

Products and coproducts also drastically differ in categories like the category of groups or rings. The product is the obvious thing, but the coproduct in the category of groups is the free product, and the coproduct in the category of rings is a ring-theoretic version of the free product. The coproduct in the category of commutative rings is the tensor product, which is also very different from the product.

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More generally, in a situation like this I think you should try to acquaint yourself with more examples. – Qiaochu Yuan May 12 '11 at 8:58
Many thanks Yuan – El Moro May 12 '11 at 9:31

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