A category \(\mathsf{C}\) with a class of admissibly morphisms generating a Grothendieck topology where
- \(\mathsf{C}\) admits finite product
- Admissibles are closed under composition, pullback, retract
- If \(g,h\) are admissible then \(f\) is admissible in:
T-structures
Leads to an analog of a ringed space: for \(\mathsf{T}\) a pregeometry and \(X\) an infty topos, a \(\mathsf{T}{\hbox{-}}\)structure on \(X\) is a functor \({\mathcal{O}}:\mathsf{T}\to X\) which
- Preserves finite products,
- Sends pullbacks of admissibles in \(\mathsf{T}\) to pullbacks in \(X\)
- Sends coverings in \(\mathsf{T}\) to effective epis in \(X\)
Idea: think of \(\mathsf{T}_{\mathrm{an}}(k){\hbox{-}}\)structure \({\mathcal{O}}\) as a sheaf of derived rings equipped with an analytic structure. Then \({\mathcal{O}}({\mathbf{A}}^1)\) is like a sheaf of simplicial commutative rings and \({\mathcal{O}}({\mathbb{D}}_1)\) is like a subsheaf of functions \(f\) with \({\left\lVert {f} \right\rVert}\leq 1\).