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

An quick introduction is in part 3, 4 of

• André Joyal, A crash course in topos theory – The big picture, lecture series at Topos à l’IHES, November 2015, Paris

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 (2005) no. 2, 257–372 doi 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 Elephant for $(\infty,1)$-topos theory is still to be written.