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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

40 votes
2 answers
3k views

Ultrafilters as a double dual

Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known: $X$ canonically embeds into $\beta X$ (by taking principal ultrafilters); If $X$ is finite, then ther …
Adam P. Goucher's user avatar
15 votes

What is an explicit bijection in combinatorics?

Here's an example (credit: Paul Russell) of the sort of bijection you want to rule out. Question: Find an explicit bijection $f$ between the size-$k$ and size-$(k + 1)$ subsets of $\{1, 2, \dots, 2k+ …
Adam P. Goucher's user avatar
7 votes

When were triples called monads for the first time?

P. T. Johnstone (who wrote several books on Topos Theory) gave a Category Theory lecture in which he said this was originally called 'the standard construction', then 'triples', and finally 'monads' - …
Adam P. Goucher's user avatar
1 vote
1 answer
251 views

Natural bijection between sets with coloured elements?

In Andreas Blass's famous paper `Seven Trees in One', the existence of a natural bijection between binary trees and 7-tuples of binary trees is related to the equation $T^7 = T$ being satisfied by a c …
Adam P. Goucher's user avatar