nLab (infinity,1)-topos theory

Idea

The theory of (∞,1)-toposes, generalizing topos theory from category theory to (∞,1)-category theory: “geometric homotopy theory”.

Related concepts

• topos theory

• 2-topos theory

• $(\infty,1)$-topos theory

• higher topos theory

• Awodey's conjecture

References

For more see the references at (infinity,1)-topos:

Lecture notes:

• Charles Rezk: Lectures on Higher Topos Theory, Leeds (2019) [pdf, pdf]

An quick introduction is in parts 3-4 of

• André Joyal: A crash course in topos theory – The big picture, lecture series at Topos à l’IHES, Paris (November 2015) [videos: part 1, 2, 3, 4]

For origins of the notion of $(\infty,1)$-topos itself see the references at (∞,1)-topos.

Early frameworks for Grothendieck (as opposed to “elementary”) $(\infty,1)$-topoi are due Charles Rezk via model categories

• Charles Rezk, Toposes and homotopy toposes, 2010 (pdf)

and due to Toën–Vezzosi in two versions (preprints 2002), via simplically enriched categories and via Segal categories:

• Bertrand Toën, Gabriele Vezzosi: Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 2 (2005) 257-372 [doi:10.1016/j.aim.2004.05.004, arXiv:math.AT/0207028]

• Bertrand Toën , Gabriele Vezzosi, Segal topoi and stacks over Segal categories, arXiv:math.AG/0212330

A general abstract conception of $(\infty,1)$-topos theory in terms of (∞,1)-category theory was given in

• Jacob Lurie, Higher Topos Theory .

The analog of the Johnstone 2002 for $(\infty,1)$-topos theory is still to be written.