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Category theory is the rigorous study of abstract systems of objects and mappings. As such it is very widely applicable, and it has become indispensable in many parts of modern mathematics, including algebra, topology, logic, and theoretical computer science.
The fragmentation of mathematics into many specialized areas makes it impossible for any individual mathematician to understand more than a small fraction of the subject. The success of category theory is due in part to its usefulness in counteracting this problem. Vague analogies between phenomena in disparate parts of mathematics can often be made precise using category theory; and having been made precise, they can be developed in a systematic way, leading one to spot new points in common that would not have been spotted otherwise. We will see many examples of this principle in action.
The main aim of the course is to show how categorical language can simplify mathematical thought and unify apparently unrelated areas of mathematics. Participants will learn this language and see examples of how it is used in various other fields, especially algebra and topology. They will also gain technical proficiency in manipulating categorical concepts.
There are no essential prerequisites. Most important is a willingness to think abstractly. Some knowledge of undergraduate algebra and topology would also be an advantage, as many of the examples will be taken from those subjects.
If time allows, we may also cover either monads or monoidal categories, according to demand.
Ten 1.5-hour classes, one per week. The course will closely follow the lecturer's book Basic Category Theory (Cambridge University Press, 2014). Participants will be expected to read the relevant section in advance of each class and do some exercises; class time will be used for further explanation, exercises, and trouble-shooting. Participants may also be requested to present material.