nLab 2-topos

Idea

The notion of 2-topos is the generalization of the notion of topos from category theory to the higher category theory of 2-categories.

There are multiple conceivable such generalizations, depending in particular on whether one tries to generalize the notion of Grothendieck topos or of elementary topos, and in the latter case what axioms one chooses to take as the basis for generalization.

In contrast, (2,1)-toposes are much better understood.

A Grothendieck 2-topos is a 2-category of 2-sheaves over a 2-site.

A Grothendieck (2,1)-topos is a (2,1)-category of (2,1)-sheaves over a (2,1)-site.

See also higher topos theory.

Properties

Characterization of 2-sheaf 2-toposes

The 2-toposes of 2-sheaves over a 2-site are special among all 2-toposes, in direct generalization of how sheaf toposes (“Grothendieck toposes”) are special among all toposes. In that case, Giraud's theorem famously characterizes sheaf toposes. This characterization has a 2-categorical analog: the 2-Giraud theorem.

$(n,r)$-Localic 2-toposes

A 2-sheaf 2-topos is “$(n,r)$-localic” or “$(n,r)$-truncated” if it has an (n,r)-site of definition.

In particular a $(2,1)$-localic 2-topos is the same as a (2,1)-topos.

In terms of internal categories

Given a 2-topos $\mathcal{X}$, regard it is a 2-site by equipping it with its canonical topology.

Definition

Write $Cat(\mathcal{X})$ for the 2-category of internal categories in $\mathcal{X}$, precisely: the 2-category of 2-congruences and internal anafunctors between them (see here).

Theorem

For $\mathcal{X}$ a 2-topos of 2-sheaves on a 2-site, there is an equivalence of 2-categories

$$\mathcal{X} \simeq Cat(\mathcal{X}) \,.$$

If $\mathcal{X}$ is $(2,1)$-localic, with a (2,1)-site of definition $C$, then there is already an equivalence

$$\mathcal{X} \simeq Cat(Sh_{(2,1)}(C))$$

with the 2-category of categories internal to the underlying (2,1)-topos.

If $\mathcal{X}$ is $1$-localic, with 1-site of definition, then there is even already an equivalence

$$\mathcal{X} \simeq Cat(Sh(C))$$

with the internal categories in the underlying sheaf topos.

Proof

By the 2-Giraud theorem, $\mathcal{X}$ is an exact 2-category. With this, the first statement is this theorem at 2-congruence.

By the discussion at n-localic 2-topos, a 2-sheaf 2-topos has enough groupoids precisely if it has a (2,1)-site of definition, and has enough discretes precisely if it has a 1-site of definition. With this the second and third statement is this theorem at 2-congruence.

Remark

The noteworthy point about theorem is that for an ambient context which is a $(2,1)$-localic (2,1)-topos, the straightforward morphisms of internal categories, hence the notion of internal functors, needs no further localization. This is in stark contrast to the situation for an ambient 1-category. The generalization of this phenomenon is discussed at category object in an (∞,1)-category.

Examples

The archetypical 2-topos

The archetypical 2-topos is Cat. This plays the role for 2-toposes as Set does for 1-toposes.

Internal categories in a $(2,1)$-topos

Given any (2,1)-topos $\mathcal{X}$, the 2-category $Cat(\mathcal{X})$ of internal categories in $\mathcal{X}$ ought to be a 2-topos. But it seems that at the moment there is no proof of this in the literature.

For literature on internal categories in 1-toposes see at 2-sheaf.

Related concepts

• elementary theory of the 2-category of categories (ETCC)

• formal category theory

flavors of higher toposes

• 1-topos

  • (0,1)-topos

  • (1,1)-topos = topos

  • (2,1)-topos

  • $(n,1)$-topos

  • (∞,1)-topos ($n$-localic)

    $\;\;\;\;$ model topos

• 2-topos

  • (2,2)-topos ($n$-localic)

  • (∞,2)-topos

• $n$-topos

  • $(\infty,n)$-topos

References

An introduction is in

• Mike Shulman, What is a 2-topos?

Early developments include

• Ross Street, Two dimensional sheaf theory, J. Pure and Appl. Algebra 24 (1982) 2Opp.

• Dominique Bourn, Sur les ditopos, C. R. Acad. Sci. Paris 279, 911–913 (1974).

A detailed discussion from the point of view of internal logic is at

• Mike Shulman, 2-categorical logic

Discussion of the 2-categorical Giraud theorem for 2-sheaf 2-toposes is in

• Ross Street, Characterization of Bicategories of Stacks Category theory (Gummersbach 1981) LNM 962, 1982, MR0682967 (84d:18006)

• Mike Shulman, 2-Giraud theorem

Discussion of the elementary topos-analog of 2-toposes is in

• Mark Weber, Yoneda structures from 2-toposes, Appl Categor Struct 15 (2007) 259–323 [doi:10.1007/s10485-007-9079-2, pdf]

A notion of “flat 2-functor” (cf Diaconescu's theorem) perhaps relevant to the “points” of 2-toposes is in

• M.E. Descotte, Eduardo Dubuc, M. Szyld, On the notion of flat 2-functors, arXiv:1610.09429

Discussion of 2-classifiers for 2-toposes is in

• Luca Mesiti, 2-classifiers via dense generators and Hofmann-Streicher universe in stacks [arXiv:2401.16900]