A category $\cat C$ with a class of admissibly morphisms generating a [Grothendieck topology](Grothendieck%20topology) where

- $\cat C$ admits finite product
- Admissibles are closed under composition, pullback, retract
- If $g,h$ are admissible then $f$ is admissible in:
  
\begin{tikzcd}
	& Y \\
	X && Z
	\arrow["f", from=2-1, to=1-2]
	\arrow["g", color={rgb,255:red,92;green,92;blue,214}, from=1-2, to=2-3]
	\arrow["h"', color={rgb,255:red,92;green,92;blue,214}, from=2-1, to=2-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwxLCJYIl0sWzIsMSwiWiJdLFsxLDAsIlkiXSxbMCwyLCJmIl0sWzIsMSwiZyIsMCx7ImNvbG91ciI6WzI0MCw2MCw2MF19LFsyNDAsNjAsNjAsMV1dLFswLDEsImgiLDIseyJjb2xvdXIiOlsyNDAsNjAsNjBdfSxbMjQwLDYwLDYwLDFdXV0=)

# T-structures

Leads to an analog of a ringed space: for $\cat T$ a [pregeometry](pregeometry.md) and $X$ an [infty topos](infty%20topos), a $\cat T\dash$structure on $X$ is a functor $\OO:\cat T\to X$ which

- Preserves finite products,
- Sends pullbacks of admissibles in $\cat T$ to pullbacks in $X$
- Sends coverings in $\cat T$ to [effective epis](effective%20epis) in $X$

Idea: think of $\cat T_\an(k)\dash$structure $\OO$ as a sheaf of [derived](Unsorted/derived%20scheme.md) rings equipped with an analytic structure.
Then $\OO(\AA^1)$ is like a sheaf of [simplicial commutative rings](Unsorted/simplicial%20commutative%20ring.md) and $\OO(\DD_1)$ is like a subsheaf of functions $f$ with $\norm{f}\leq 1$.