Algebraic and combinatorial aspects of category theory




Abstract

Category theory, invented around 1945, has demonstrated a certain efficiency and unifying power in some areas of mathematics like algebraic geometry, algebraic topology, logic, and computer science.
Although it is often presented as a language for compositionality, that is, for describing how systems can be built out of simpler parts, this course will focus on perhaps less well known, yet basic facets of the theory.
Namely, we will introduce categorical toolkits for dealing with algebraic structure and combinatorics. Time permitting, we will also explain how category theory relates the algebraic and combinatorial worlds:
  • it automatically equips algebraic structures with a combinatorial presentation, and
  • it automatically promotes certain classes of combinatorial structures as algebraic ones.
Part of the fun lies in defining “combinatorial” and “algebraic” structures in the first place, which category theory does rather efficiently.


Syllabus

Basic concepts

  • Categories, functors, natural transformations
  • Limits and colimits

Algebraic structure

  • Monads
  • Adjunctions
  • Monadicity
  • Free monads

Combinatorial structure

  • Presheaves
  • Species of structure
  • Factorisation systems
  • Cofibrant generation

The combinatorial structure of algebras, and vice versa

  • Nerves of monads
  • Monads induced by factorisation systems
  • Locally presentable categories


References

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    Free online version.
  • Adámek and Rosický. Locally Presentable and Accessible Categories. London Mathematical Society Lecture Note Series 189. Cambridge University Press, Cambridge, 2014.
  • Leinster. Basic Category Theory. Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, Cambridge, 2014.
    Free online version.
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    London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1982.
    Free TAC reprint.
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    Free online version.
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    Free online version.
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    Free online version.