Formal moduli via fractal infinity categories

defn - fractal infinity category

Definition A pointed $\infty$-category $C$ admitting finite products is said to be fractal if the functor

$Aut:C\to Gp(C)$ is an equivalence.

Remark

• This says 0-groups coincide with 1-groups, hence the fully invertible part of the categorification tower is constant.

Examples

• the 0-category (0-object in pointed categories) is fractal
• $\infty$-categories of formal moduli problems are fractal.

Construction of Aut

Mimicking the construction in constr - unpointed toposic Koszul duality. This was also constructed in the work of Hoyois-Safranov-Schrotzeke-Sibilla on categorified traces.

Link to original

defn - (formal) moduli-objects in a pointed category

Definition Let $C$ be a pointed $\infty$-category admitting finite products, the category of moduli objects of $C$, $C^{\%}$ is the largest fractal subcategory (defn - fractal infinity category) containing the basepoint.

Example Since $\{0\}$ is fractal, every pointed category (maybe we also need $C$ to contain sequences of subcategories) has a moduli completion.

Example (\infty-topoi) In particular, $Anim{a}_{\ast }^{\%}\simeq \{0\}\subset Anim{a}_{\ast }$, this tells us that for any $\infty$-topos $\mathfrak{X}$, the global sections functor ${\mathfrak{X}}_{\ast }\to Anim{a}_{\ast }$ loses all data about the moduli completion ${\mathfrak{X}}_{\ast }^{\%}$. Hence, moduli completion only adds “local data”. Conversely, that $Anima$ has no trivial moduli-objects can be seen as recording the fact that sheaves on a point has no local data.

Example (Moduli completions) defn - moduli-completion of a category

Link to original

defn - moduli-completion of a category

Definition The moduli-completion of an $\infty$-category in $C$ is given by

${\hat{C}}_{moduli}:=P(C{)}_{\ast }^{\%}$ (here the RHS is moduli completion of pointed presheaves, see defn - (formal) moduli-objects in a pointed category)

Project/conjecture

Let $\mathcal{O}$ be a non-unital $\infty$-operad, $V^{\otimes }$ a presentably symmetric monoidal stable $\infty$-category. Let $D\mathcal{O}$ denote the Koszul dual operad. There’s an equivalence

$${\widehat{Al{g}_{\mathcal{O}}(V)}}_{moduli}\simeq Al{g}_{D\mathcal{O}}(V)$$

That is, algebras over $\mathcal{O}$ moduli-complete to algebras over $D\mathcal{O}$.

Remark: even for $\mathcal{O}\in \{Comm,E_k\}$, the LHS above generalizes the category of formal moduli constructed by Lurie, Gatisgory-Rozenblyum in several ways

• It makes sense for $V\ne Mo{d}_{\mathbb{C}}$. Of particular interest are examples
  • $V=Mo{d}_R(Sp)$ for $R$ an $E_{\infty }$ ring spectrum. For example, $R\in \{\mathbb{S},T(n),K(n),TMF,H\mathbb{Z}/p^n\}$
  • It might contain “curved deformations/formal moduli equipped with (non-flat connections/twisted D-modules)“.

Also, we expect ${\widehat{Al{g}_{\mathcal{O}}(V)}}_{moduli}$ to admit various full subcategories constructed using a generalization of the notion in defn - nilpotent maps of commutative ring spectra

Link to original

defn - functorial algebra tower

Definition: A functorial tower of algebras $\rho$ is the data of

• a functor

$${\rho }_{\bullet }:CAl{g}_{nu}\to Fun({\mathbb{Z}}^{\ge },CAl{g}_{nu})$$

• a natural transformation $T_{\rho }:\Delta \to {\rho }_{\bullet }$

i.e. it’s the data for each $I\in CAl{g}_{nu}$ a tower of non-unital $E_{\infty }$ algebras under $I$, functorial in $I$.

Examples include $E_n$ Goodwillie towers for all $n$, the Cech/Amitsur filtration, and our conjectured $E_n$ Amitur filtrations.

Definition: An algebra $I\in CAl{g}_{nu}$ is said to be $\rho$-complete if the map

$$I\to \underset{n}{\lim }{\rho }_n(I):={\rho }_{\infty }(I)$$

is an equivalence.

Generalizations: This obviously generalizes with $CAl{g}_{nu}$ replaced by an arbitrary $\infty$-category.

Link to original

defn - nilpotent maps of commutative ring spectra

Let $f:R\to A$ be a map of $E_{\infty }$ ring spectra. The kernel of the underlying $R$-module map $Fib(\underline{f})$ lifts to a non-unital $E_{\infty }$ algebra, we denote this object by $I_f\in CAl{g}_{nu}(Mo{d}_R)$.

Let $\rho$ be a functorial tower of algebras, as in defn - functorial algebra tower.

Definition

$f$ is said to be $\rho$-nilpotent if $I_f$ is $\rho$-complete.

In the following definitions we’ll fix a $\rho$ omit it from the notation.

Definition The category of formal $R$-algebras, $\widehat{CAl{g}_R}\subset CAl{g}_R$ is full subcategory spanned by maps $u:R\to A$ such that $u$ is $\rho$-nilponent.

Definition The category of formal extensions of $A$, $\widehat{Ext{n}_A}\subset CAl{g}_{/A}$ is full subcategory spanned by maps $p:B\to A$ such that $p$ is $\rho$-nilponent.

Link to original

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

Link to original