Background

Let $C$ be an (extensive) category. An object $X\in C$ is called connected if the functor $Hom(X,-):C\rightarrow Set$ preserves coproducts.

Given a category $C$, one can consider the category $Fam(C)$ of set-indexed familiies of objects in $C$. Formally, the objects are pairs $(X,F)$ in which $X$ is a set and $F:X\rightarrow C$ is a functor and a map $(X,F)\rightarrow (Y,G)$ is pair $(f,g)$ where $f:X\rightarrow Y$ is a function and $g:F\Rightarrow G\circ f$ is a natural transformation. The connected objects of $Fam(C)$ are precisely the singleton families, i.e pairs $(X,F)$ where $X$ is a one-element set.

The notion of an object being connected should have a strightforward generalization to the $(\infty,1)$-categorical case. More precisely, call an object $X$ of an $(\infty,1)$-category $C$ connected if $Hom(X,-):C\rightarrow Grpd_{\infty}$ preserves coproducts.

Additionally, $Fam(C)$ also has an analogous construction for $C$ an $(\infty,1)$-category. Instead of looking at set-indexed families, we consider families of objects indexed by $\infty$-groupoids. So the objects of $Fam(C)$ are pairs $(X,F)$ where $X$ is an $\infty$-groupoid and $F:X\rightarrow C$ is a functor. The morphisms are exactly the same as the 1-categorical case.

Question

What are the connected objects of $Fam(C)$ when $C$ is an $(\infty,1)$-category? I think that this is the case when $X$ is a connected $\infty$-groupoid.

Background

Let $C$ be an (extensive) category. An object $X\in C$ is called connected if the functor $Hom(X,-):C\rightarrow Set$ preserves coproducts.

Given a category $C$, one can consider the category $Fam(C)$ of set-indexed familiies of objects in $C$. Formally, the objects are pairs $(X,F)$ in which $X$ is a set and $F:X\rightarrow C$ is a functor and a map $(X,F)\rightarrow (Y,G)$ is pair $(f,g)$ where $f:X\rightarrow Y$ is a function and $g:F\Rightarrow G\circ f$ is a natural transformation. The connected objects of $Fam(C)$ are precisely the singleton families, i.e pairs $(X,F)$ where $X$ is a one-element set.

The notion of an object being connected should have a straightforward generalization to the $(\infty,1)$-categorical case. More precisely, call an object $X$ of an $(\infty,1)$-category $C$ connected if $Hom(X,-):C\rightarrow Grpd_{\infty}$ preserves coproducts.

Additionally, $Fam(C)$ also has an analogous construction for $C$ an $(\infty,1)$-category. Instead of looking at set-indexed families, we consider families of objects indexed by $\infty$-groupoids. So the objects of $Fam(C)$ are pairs $(X,F)$ where $X$ is an $\infty$-groupoid and $F:X\rightarrow C$ is a functor. The morphisms are exactly the same as the 1-categorical case.

Question

What are the connected objects of $Fam(C)$ when $C$ is an $(\infty,1)$-category? I think that this is the case when $X$ is a connected $\infty$-groupoid.