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Mathematics > Category Theory

arXiv:1912.10642 (math)
[Submitted on 23 Dec 2019 (v1), last revised 18 Apr 2024 (this version, v7)]

Title:Notes on Category Theory with examples from basic mathematics

Authors:Paolo Perrone
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Abstract:These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor any of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. The only knowledge that is assumed from the reader is linear algebra. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept. The reader is encouraged to read all the examples, this way they may even learn something new about a different field.
Particular emphasis is given to the Yoneda lemma and its significance, with both intuitive explanations, detailed proofs, and specific examples. Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. From the applied point of view, this shows why categorical thinking can help whenever some process is taking place on a graph. From the pure math point of view, this can be seen as the 1-dimensional first step into the theory of simplicial sets. Finally, monads and comonads are treated on an equal footing, differently to most literature in which comonads are often overlooked as "just the dual to monads". Theorems, interpretations and concrete examples are given for monads as well as for comonads.
This work, thoroughly revised and expanded, is now a book, with an extra section on monoidal categories.
Comments: Lecture notes, 181 pages
Subjects: Category Theory (math.CT); Logic in Computer Science (cs.LO); History and Overview (math.HO)
Cite as: arXiv:1912.10642 [math.CT]
  (or arXiv:1912.10642v7 [math.CT] for this version)
  https://doi.org/10.48550/arXiv.1912.10642
arXiv-issued DOI via DataCite
Journal reference: Starting Category Theory, World Scientific, 2024
Related DOI: https://doi.org/10.1142/13670
DOI(s) linking to related resources

Submission history

From: Paolo Perrone [view email]
[v1] Mon, 23 Dec 2019 06:38:45 UTC (243 KB)
[v2] Sun, 5 Jan 2020 02:26:10 UTC (243 KB)
[v3] Tue, 17 Mar 2020 16:27:21 UTC (243 KB)
[v4] Tue, 2 Jun 2020 15:53:30 UTC (243 KB)
[v5] Tue, 25 Aug 2020 13:46:09 UTC (243 KB)
[v6] Tue, 9 Feb 2021 16:39:48 UTC (243 KB)
[v7] Thu, 18 Apr 2024 21:36:09 UTC (198 KB)
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