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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?

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    $\begingroup$ Do you have something in mind in the 1-categorical case? Like a comparison between categories opfibered over O_G^{op} and categories enriched in presheaves of sets on the orbit category? My first guess is that these things (in the 1-categorical and infty-categorical cases) should be quite different. Maybe categories internal to G-spaces gets you closer? But they still feel different. $\endgroup$ – Dylan Wilson Jun 10 '18 at 23:30
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    $\begingroup$ From an ∞-category enriched in presheaves on $C$ you should get a presheaf of ∞-categories on $C$ by changing the enrichment via the evaluation functors. This construction is a right adjoint functor, and naively I would expect it to be fully faithful but not essentially surjective. $\endgroup$ – Marc Hoyois Jun 11 '18 at 2:48
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    $\begingroup$ Yeah, not all objects over $G/e$ will come by restriction from an object over $G/G$ in a $G$-parametrized category (silly example: take $EG$ as a $G$-parametrized groupoid). I think you might have some hope for full faithfulness though $\endgroup$ – Denis Nardin Jun 11 '18 at 6:23
  • $\begingroup$ Okay, maybe the better comparison is to categories internal to $Top_G$. $\endgroup$ – Tim Campion Jun 11 '18 at 12:56
  • $\begingroup$ @DylanWilson I think the edit gives an idea how the comparison should go. I wasn't sure before. $\endgroup$ – Tim Campion Jun 11 '18 at 13:35
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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?

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  • $\begingroup$ All of this works for an arbitrary presheaf category. That is, if $T$ is any small $\infty$-category, then categories fibered over $T$ are the same as categories internal to presheaves on $T$. That is, so long as you agree with my formulation of univalence for internal categories -- which admittedly doesn't necessarily make sense if you want to work internal to an arbitrary finitely-complete category. But at least it makes sense in the presheaf case. I suppose if you wanted it to work for an arbitrary finitely-complete category, you could use an appropriate truncation of $E_\bullet$. $\endgroup$ – Tim Campion Jun 11 '18 at 14:38
  • $\begingroup$ This leaves me wondering how much of the parameterized category theory developed by Barwick et al can be recast as general internal category theory. $\endgroup$ – Tim Campion Jun 11 '18 at 14:57
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    $\begingroup$ I think you can get the statement you want for enriched categories from a variant of the proof of theorem 4.4.6 of Gepner and Haugseng enriched ∞-categories paper $\endgroup$ – Denis Nardin Jun 11 '18 at 15:49
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    $\begingroup$ It looks like some discussion of this in the 1-categorical context is in Johnstone-Pare's book on "indexed categories and their applications" page 25. (Their "indexed categories" are basically the 1-categorical versions of today's "parameterized categories" ) $\endgroup$ – Dylan Wilson Jun 11 '18 at 16:49
  • $\begingroup$ (sorry, should be Johnstone-Pare-Rosebrugh-Schumacher-Wood-Wraith, and the chapter in question is by Pare-Schumacher) $\endgroup$ – Dylan Wilson Jun 11 '18 at 16:50

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