nLab logos

The term “logos” is used

• by Heraclitus for logos (in philosophy), something that later was discussed as “logic” in Hegel‘s Science of Logic.

• in Freyd-Scedrov as a synonym for Heyting category;

• in Joyal 08 as a synonym for quasi-category ((∞,1)-category, cf. also Joyal locus);

• in Joyal 2015, Anel & Joyal 2019, ABFJ 2023 p. 3 for the formal dual of a topos, namely for an object of the opposite of the category of toposes. This is along the same lines as the classical formal duality between frames and locales ((0,1)-toposes) and extends immediately to $(\infty,1)$-toposes whose formal duals are hence “$(\infty,1)$-logoi” or “$\infty$-logoi”, for short;

• in Anel 2019 for a cartesian closed $\infty$-category with finite $\infty$-limits and van Kampen colimits in size bounded by some inaccessible cardinal.

• in DLL24 for a cocomplete category equipped with a family of limits such that colimits are exact with respect to this family. This paper also discusses the relationship between theirs, Anel-Joyal and Freyd-Scedrov proposal of logos. Despite being different these proposals all come with similar motivations in mind.

References

• Peter Freyd, Andre Scedrov, Categories, Allegories

• André Joyal, Notes on Logoi, 2008 (pdf)

• André Joyal, A crash course in topos theory : the big picture, lecture series at Topos à l’IHÉS, 2015 Video 1, Video 2, Video 3, Video 4.

• Mathieu Anel, André Joyal, Topo-logie, in New Spaces for Mathematics and Physics, Cambridge University Press (2021) 155-257 [doi:10.1017/9781108854429.007, pdf]

• Mathieu Anel, Descent and Univalence, talk at HoTTEST (May 2019) [slides, video]

• Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, Left-exact Localizations of ∞-Topoi III: The Acyclic Product [arXiv:2308.15573]

• Ivan Di Liberti, Gabriele Lobbia, Sketches and Classifying Logoi [arXiv:2403.09264]

Discussion of relation to taoism?:

• Elena Butti, A comparison between Heraclitus’ “Logos” and Lao-Tzu’s “Tao”, Ephemeris (2013) [pdf]