After having completed this course, the student should:
* know the principles of category theory
* have seen a large number of examples of categories and functors
* be able to apply the basic principles of category theory in new settings
* be familiar with modules over a ring
* be able to analyse the structure of modules over some particular classes of rings, such as PIDs
* be able to work with free resolutions
* be familiar with the first principle of homological algebra
* be able to carry out calculations with complexes, homotopies of complexes and the calculation of derived functors. |
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In this course we will give a first introduction to category theory, aimed at mathematicians. This provides us with a uniform langage that is useful to analyse _structures_ regardless of in which branch of maths they appear. As such it is an indispensible tool for further study of pure maths.
In parallel with this we will develop some Algebra, notably the theory of modules over rings. This is of great independent interest but also serves to illustrate many of the categorical notions that we will see.
The two strands are interwoven in the second half of the course, when we will give a first introduction to homological algebra. Here begins a mathematical development that has shaped much mathematics since the 1950s and today is an indispensible tool in many branches of maths.
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