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