Category Archives: Vignettes.

Vignettes. Part 13.

There probably is no better illustration of the vagaries of notational fashion than the fact that nowadays, the numeral 1 is commonly used to denote the (conceptually very important) terminal category, while George Boole 170 years ago used 1 to denote … Continue reading

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Vignettes. Part 10.

In one of his very useful writings, Robert Harper writes the following thoughtful sentence: But the upshot of Gödel’s Theorem is that as soon as we fix the concept of formal proof, it is immediate that it is not an … Continue reading

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Vignettes. Part 8.

The following is an illustration of a very widely known, yet not widely-enough-known basic connection between things. Explanations will not be given. They can easily be found, given internet access. This connection gives a structure and systematic explanation for some … Continue reading

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Vignettes. Part 7.

Did you know that mathematicians have beautifully imaginative ways to conceive of lines as cycles? For example, for many, is just a string of symbols. Yet, e.g. for mathematicians working in homological algebra this is kind-of-a-cycle in the sense that … Continue reading

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Vignettes. Part 6.

If one cares for categories, one should intentionally take an intensional view on them:  axioms matter. In particular, associativity is an axiom. But it is excusable that one sometimes forgets that associativity is axiomatic: so many categories are concretizable (and then, … Continue reading

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Vignettes. Part 1. Complete Graphs via right-adjoints to forgetting.

The following is widely-, though not widely-enough-known. Let a category of graphs of your choice. Probably any reasonable category of graphs will work for you, mutatis mutandis, as they say, provided that the morphisms of are (at least easily interpretable … Continue reading

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Vignettes. Part 0. An observation.

The Four Color Theorem is about (not-so-constructively) proving an upper bound. The Map Color Theorem is about (constructively) proving a (parametrized) lower bound, and its name, adorning a nice monograph on the subject, is, as already Daniel Kleitman pointed out … Continue reading

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