Summary of short course and other notes

The point-free approach to sheafification

Abstract: Let $\Omega$ be an arbitrary frame, and consider the category $\Psh(\Omega)$ of presheaves on $\Omega$ and the and the category $\Set(\Omega)$ of $\Omega$-sets. There are various other associated categories and functors between them. I describe these with emphasis on the functor which separates a presheaf and the functor that sheafifies a presheaf.

49 pages + 13 pages of solutions

Full notes from omegasets.dvi.gz or omegasets.ps.gz or omegasets.pdf.gz .



Forms of recursion and induction

Abstract: A brief introduction to subrecursive hierarchies and related topics, but not intended as a detailed account. The notes deal with such topics as the various kinds of recursions (over the natural numbers), the primitive recursive functions and the way these are stratified, the complexities beyond primitive recursion as can be generated by jump operators.

60 pages + 17 pages of solutions

Full notes from FormsRec.dvi.gz or FormsRec.ps.gz or FormsRec.pdf.gz .



$\lambda$-calculi

Abstract: An introduction to the untyped, and the simply typed $\lambda$-calculus, and G\"odel's T as a $\lambda$-calculus. The emphasis is on the way natural number gadgets are simulated amd represented in the various calculi.

46 pages + 12 pages of solutions

Full notes from lcalculus.dvi.gz or lcalculus.ps.gz or lcalculus.pdf.gz .



While loops and programs

Abstract: An short introduction to the denotational semantics of While loops.

29 pages + 6 pages + 8 pages of solutions

Full notes from While.dvi.gz or While.ps.gz or While.pdf.gz .



Category theory

Abstract: An introduction to category theory in four chapters dealing with the basics, functors and natural transformations, various kinds of limits and universal solutions, and cartesian closed categories. Contains many exercises (and eventually will contain a fairly full set of solutions). The best introduction to the subject ever written.

133 pages + solutions to be written

Full notes from CatTheory.dvi.gz or CatTheory.ps.gz or CatTheory.pdf.gz .



Domains for recursion

Abstract: A short introduction to domains from various forms of recursion over the natural numbers.

19 pages + 14 pages of solutions

Full notes from Domains.dvi.gz or Domains.ps.gz or Domains.pdf.gz .



The coverage technique for enriched posets

Abstract: The coverage technique is used to convert a poset (perhaps with some extra structure) into a complete poset of a certain kind, usually a frame or quantale. These notes describe the basics of the method with a decent selection of examples. Especially written for the hard of hearing.

84 pages

Full notes from Coverages.dvi.gz or Coverages.ps.gz or Coverages.pdf.gz .

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Harold Simmons
Last modified: Monday 13 January 2003