constr - unpointed toposic Koszul duality

Let $\mathfrak{X}$ be an $\infty$-topos, $X$ and object. We construct an adjunction

Where the bottom category is defn - Symtr(X, C), category of symmetries of an object

Construction

$\omega$ takes $f:X\to Y$ to the symmetry given by $\check{c}(f)$.

$\beta$ takes an algebra $A$ acting on $X$ to the geometric realization of the simplicial object

$Ba{r}_{{\mathfrak{X}}_{/X}}^{\bullet }(G,G,X)$

The map $X\to \beta (A)$ is given by including the 0-th term in the simplicial object.

Digaram https://q.uiver.app/?q=WzAsMixbMCwwLCJcXG1hdGhmcmFrIFhfe1gvfSJdLFswLDEsIlN5bXRyKFgsIFxcbWF0aGZyYWsgWCkiXSxbMCwxLCJcXG9tZWdhIiwwLHsibGFiZWxfcG9zaXRpb24iOjQwLCJvZmZzZXQiOi00fV0sWzEsMCwiXFxiZXRhIiwwLHsib2Zmc2V0IjotNH1dLFszLDIsIiIsMix7ImxldmVsIjoxLCJzdHlsZSI6eyJuYW1lIjoiYWRqdW5jdGlvbiJ9fV1d