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By Bergman G.M.

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Extra info for A companion to S.Lang's Algebra 4ed.

Example text

Different authors make different choices. 160-161) uses in both constructions the same sort of ordered set, called a ‘‘directed’’ partially ordered set, and indexes the directed system in different ways in the two cases. The background of this choice is that the earliest cases that were considered, before the general concept was developed, involved ‘‘sequences’’ of groups, and people naturally indexed both the directed and inverse systems by the positive integers. However, be prepared to see different notations in different works.

11:6(a). Category theory and set theory (optional). If one wants to study concepts like ‘‘the category of all groups’’, one is faced with the problem that under conventional set theory, all groups form a class, but not a set, and there is very little one is allowed to do with classes. One often calls a category which forms a genuine set a ‘‘small category’’, and though one can work easily with these, they exclude most of the categories one is interested in. Here is a creative way out of this dilemma, due to Alexander Grothendieck.

Then we say that G is the group presented by the generators S and relations T if the relations constituting T imply all relations that hold among the elements of the generating set S. This is equivalent to saying that G is the initial object (‘‘universal repelling object’’) in the category of groups given with S-tuples of elements satisfying the system of relations T. For example, our characterization of the group Zn in our ‘‘Further notes on functors’’ above is equivalent to saying that it can be presented by one generator x and one relation, x n = e.

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A companion to S.Lang's Algebra 4ed. by Bergman G.M.

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