nLab (1,2)-topos

Idea

The notion of (1,2)-topos should be the notion of higher toposes among (1,2)-categories or 2-posets.

Examples

Example

The (1,2)-category Pos of posets and monotone maps should be the archetypal $(1,2)$-topos.

The poset of truth values

$$\big( \bot \to \top \big) \;\in\; Pos$$

should play the role of the “sub-poset classifier” in $Pos$, the (1,2)-analog of the subobject classifier in a 1-topos.

Here, morphisms into it classify monic fibrations of posets, namely sieves (e.g. Exp. 9.26 here)

Related concepts

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