arXiv:2204.00948math.HO
Exploring mathematical objects from custom-tailored mathematical universes
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Abstract
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are toposes in which the axiom of choice and the intermediate value theorem from undergraduate calculus fail. The purpose of this contribution is to give a glimpse of the toposophic landscape, presenting several specific toposes and exploring their peculiar properties, and to explicate how toposes provide distinct lenses through which the usual mathematical objects of the standard topos can be viewed.
Basic results that are useful on a day-to-day basis: 1. Yoneda's Lemma, and the corollary that objects defined by universal property are unique up to unique isomorphism 2. Right adjoints preserve limits and more generally, Kan extensions The "big" theorem of basic category…
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