nLab arrow (∞,1)-topos

Definition

For $\mathbf{H}$ an (∞,1)-topos, its arrow $(\infty,1)$-topos is its arrow (∞,1)-category.

Properties

For $(X_{def} \stackrel{\iota_X}{\to} X_{bulk})$ and $(A_{def} \stackrel{\mathbf{F}}{\to} A_{bulk})$ in $\mathbf{H}^{\Delta^1}$, there is an (∞,1)-pullback diagram

$$\array{ \mathbf{H}^{\Delta^1}(\iota_X, \mathbf{F}) &\stackrel{}{\to}& \mathbf{H}(X_{def}, A_{def}) \\ \downarrow &\swArrow& \downarrow \\ \mathbf{H}(X_{bulk}, A_{bulk}) &\to& \mathbf{H}(X_{def}, A_{bulk}) } \,.$$

Note: ‘def’ and ‘bulk’ indicate terms of fields in a QFT with defects.

Related concepts

• slice (∞,1)-topos

• relative cohomology

• Sierpinski topos