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A topos is a category that behaves very much like the category of sets and possesses a good notion of localization. Related to topos are: sheaves, presheaves, descent, stacks, localization,...

6 votes

What's an example of a locally presentable category "in nature" that's not $\aleph_0$-locall...

The category of $\Delta$-generated spaces (the full category of the category of general topological spaces which are colimits of simplices) is locally $\lambda$-presentable for a regular cardinal $\la …
Philippe Gaucher's user avatar