What is the relationship between $O_G$-parameterized $\infty$-categories and $\infty$-categories enriched in $Top_G$?

Question

Let $G$ be a finite group. Barwick et al define a $G-\infty$-category to be a fibration over the orbit category $O_G$ of transitive $G$-sets. But in the non-$\infty$-land, the natural guess at where I should work to do $G$-equivariant homotopy theory is a category enriched in the category $Top_G$ of $G$-spaces. In the $\infty$ world, this setting can be generalized directly using Gepner and Haugseng's notion of enriched $\infty$-category.

Question: Is there a comparison functor between $\infty$-categories fibered over $O_G$ and $\infty$-categories enriched in $Top_G$? Is this an equivalence, perhaps after passing to certain subcategories?

EDIT: Maybe this is what Marc is driving at in the comments, but think of it this way. A category fibered over $O_G$ is a functor $O_G^{op} \to Cat$, which is a functor $O_G^{op} \times \Delta^{op} \to Top$ satisfying some conditions. A category internal to $Top_G = Fun(O_G^{op}, Top)$ is a simplicial object in $Top_G$, i.e. a functor $\Delta^{op}\times O_G^{op} \to Top$, satisfying some conditions. This leads me to post a

Revised Question: Are categories fibered over $O_G$ the same thing as categories internal to $Top_G$? Which ones correspond to enriched categories?

Answer 1

It seems that the answer to the first part of the revised question is yes: categories parameterized over $O_G$ are the same as categories internal to $Top_G$.

Let's think through this carefully. A category object should satisfy Segal conditions and a univalence (aka completeness in the sense of Rezk) condition. Admittedly the latter might not be entirely standardized for internal categories, but let's pick a formulation and run with it:

• A category parameterized over $O_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ which is, levelwise in $O_G$, a complete Segal space. So we have

  1. Segal conditions: $X(G/H,[n]) \to X(G/H,[1]) \times_{X(G/H,[0])} \dots \times_{X(G/H,[0])} X(G/H,[1])$ is an equivalence.

  2. Univalence: $X(G/H,[0]) \to Map(E_\bullet,X(G/H,\bullet))$ is an equivalence of spaces.

Here $E_\bullet$ is the (nerve of the) walking isomorphism, and $Map$ the $Top$-enriched homset of simplicial spaces.

• A category internal to $Top_G$ is a functor $X: O_G^{op} \times \Delta^{op} \to Top$ satisfying

  1. Segal conditions: $X(-,[n]) \to X(-,[1]) \times_{X(G/H,[0])} \dots \times_{X(-,[0])} X(-,[1])$ is an equivalence.

  2. Univalence: $X(-,[0]) \to GMap(E_\bullet, X(-,\bullet))$ is an equivalence of $G$-spaces.

Here $E_\bullet$ is included with trivial $G$-action, and $GMap$ denotes the $Top_G$-enriched homset of simplicial $G$-spaces.

Comparison:

These two notions are indeed equivalent. We see this as follows.

1. Since limits in a presheaf category are computed levelwise, the Segal conditions in the two cases are equivalent.

2. $GMap(E_\bullet, X(-,\bullet))(G/H) = Map(E_\bullet, X(G/H,\bullet))$ because for each $[n]$, $E(G/H,[n])$ is constant as a function of $G/H$ (for me, this implication requires a short calculation). Thus the univalence conditions are also equivalent.

Enriched categories? So I guess that a $G$-enriched category is an internal category to $GTop$ with trivial $G$-action on the space of objects? This should translate to a parameterized category where the "core" -- the maximal subfibration which is fibered in groupoids -- is constant?