Twisted-arrow construction for 2-categories

I've been looking over Lurie's DAG X, and he introduces a combinatorial construction called the twisted arrow construction for simplicial sets that generalizes the following ordinary categorical notion:

The Yoneda embedding $\mathcal{C}\to \widehat{\mathcal{C}}$ is adjunct under the hom-tensor adjunction to the functor $$\operatorname{Hom}:\mathcal{C}^{\operatorname{op}}\times\mathcal{C}\to \operatorname{Set}$$. Composing $\operatorname{Hom}$ with the inclusion of $\iota:\operatorname{Set}\hookrightarrow\operatorname{Cat}$ gives us a functor to $\operatorname{Cat}$, and to this functor we can apply the Grothendieck construction to produce a discrete fibration $$\pi:\int^{\mathcal{C}^{\operatorname{op}}\times\mathcal{C}} \operatorname{Hom} \to \mathcal{C}^{\operatorname{op}}\times\mathcal{C}$$

We call the total space of this fibration $\operatorname{Tw}(\mathcal{C})$, the twisted arrow category.

In chapter 4 of DAG X, Lurie gives a concrete and combinatorial description of how to generalize this to any simplicial set $X$ by constructing it as the pullback $\sigma^\ast X$ where $$\sigma: \Delta \to \Delta$$ sends $$[n] \mapsto [n]^{\operatorname{op}} \boxplus [n]$$ where $\boxplus$ is the ordinal sum/join (the faces and degeneracies map to the arrows in a canonical way that is induced by the join and the $\operatorname{op}$, so I am omitting them for the sake of brevity).

In order to understand things better in the case of a 2-category and maybe come up with a combinatorial description to extend to higher categories, I'm wondering if anyone has seen or could describe that $2$-grothendieck construction assodciated with the $\operatorname{Hom}$ functor in the case of strict $2$-categories. I'm curious if maybe something like just the (horizontal) join could possibly work to generalize it to the case of $\Theta_2$, which plays the role in $2$-categories that $\Delta$ plays in $1$-categories.

Thanks!

• The general Grothendieck construction for 2-categories is described in arxiv.org/abs/1212.6283 . Do you want something more specific than that? Aug 24 '17 at 23:05
• @Mike I'll give it a look up and down and let you know. I'm maybe looking for any more explicit computations of the Grothendieck construction applied to Hom in the 2-categorical case. Aug 25 '17 at 3:31
• I'd say that the 2-dimensional Grothendieck construction for $F : {\cal C} \to {\bf Cat}$ is some sort of 2-coend like $$\oint^c {\cal C}_{/\!/c} \times Fc$$ Aug 25 '17 at 16:56
• @MikeShulman I guess I'm asking if anyone has done a somewhat concrete computation describing the shape of the fibration associated with the 2-Yoneda embedding, which should be simpler than the general case, since all of the vertical cells should be 1-cells and 0-cells because the fibres are 1-categories. For example, a worked-out computation for the case of the fibration associated with the 2-Yoneda embedding of the 2-globe Aug 25 '17 at 18:30

While it is more explicit and combinatorial, this probably isn't precisely what you're looking for. It does seem relevant to your question, though. A year or so ago I worked out a 2-categorical analogue of the $(\infty,1)$ twisted arrow construction (I'm working with $(\infty,1)$-categories arising as Joyal fibrant replacements of the Duskin nerves of 2-categories). I initially tried the same approach Mike Schulman suggested in the comments, i.e., using Buckley's 2-categorical Grothendieck construction, but I found that a simpler construction was sufficient for my purposes. It's a quite straightforward generalization of the commutative (up to natural isomorphism) diagram $$\require{AMScd} \begin{CD} \operatorname{Cat} @>{N}>> \operatorname{Set}_\Delta \\ @V{Tw}VV @VV{Tw}V \\ \operatorname{Cat} @>>{N}> \operatorname{Set}_\Delta \end{CD}$$ to 2-categories. The construction is a little ad-hoc, but the basic idea is the following: From a 2-category $\mathscr{C}$, construct a new 2-category $Tw_2(\mathscr{C})$ by letting

• Objects of $Tw_2(\mathscr{C})$ are morphisms of $\mathscr{C}$
• 1-morphisms in $Tw_2(\mathscr{C})$ from $f$ to $f^\prime$ consist of diagrams $$\begin{array}{c c c } A & \overset{f}{\rightarrow} & B \\ h \downarrow\hspace{6pt} & \searrow & \hspace{6pt}\uparrow k \\ A^\prime & \underset{f^\prime}{\to} & B^\prime\\ \end{array}$$ and $$\begin{array}{c c c } A & \overset{f}{\rightarrow} & B \\ h \downarrow\hspace{6pt} & \nearrow& \hspace{6pt}\uparrow k \\ A^\prime & \underset{f^\prime}{\to} & B^\prime\\ \end{array}$$ where each triangle commutes up to a (not nec. invertible) 2-morphism, and the 2-morphisms satisfy the obvious commutativity condition. (Note that, by definition, these are the 3-simplices of the Duskin nerve of $\mathscr{C}$.)

• 2-morphisms are given by (appropriately coherent) natural transformations of diagrams which are the identity on $f$ and $f^\prime$.

The resulting category $Tw_2(\mathscr{C})$ is only lax unital. However, with the appropriate nerve constructions, the diagram $$\require{AMScd} \begin{CD} \operatorname{2Cat} @>{N}>> \operatorname{Set}_\Delta \\ @V{Tw_2}VV @VV{Tw}V \\ \operatorname{Lax2Cat} @>>{N}> \operatorname{Set}_\Delta \end{CD}$$ commutes up to natural isomorphism.

There is an obvious functor $$Tw_2(\mathscr{C}) \to \mathscr{C}\times \mathscr{C}^{\operatorname{op}},$$ though I haven't looked at its fibrancy properties yet.

Disclaimer: This is my first answer here, please let me know if I have ignored some protocol out of inexperience.

• I sort of figured out what the missing piece is, but it's extremely complicated and involves a lax version of the join for 2-categories. It doesn't have a nice and simple description combinatorial description in the globular/cellular context, but it can be described very easily in the context of Verity's complicial sets. This is close enough to what I wanted though, and I am going to accept this answer. Jan 12 '18 at 4:27
• Thanks! I'd actually be quite interested in seeing the complicial version. Have you written it up somewhere? Jan 12 '18 at 4:45
• I have only spoken with Dominic in person about it, and he said that the coming papers on Yoneda for complicial sets will be finally written up after he and Emily Riehl finish their current series of papers and book project. What he told me in person was, to paraphrase, "do the comprehension construction, but with complicial cosmoi, the lax Gray tensor product, lax join, and lax homotopy-coherent nerve". Jan 12 '18 at 6:01
• Comprehension is an alternative to twisted arrows, but twisted arrows can be done with the lax join/overcategory as an equivalent model for higher lax commas. You still need the lax hc-nerve and lax tensor product to get the "complicial set of complicial sets". Jan 12 '18 at 6:04
• There are also (very) partial results in the globular/cellular case coming from the work on lax tensors and joins of strict omega categories by Ara and Maltsiniotis, but the combinatorics is much much harder. If you're interested in this, please e-mail me. I'm looking for someone who might be interested in hearing how things fit together. Jan 12 '18 at 6:10

Since you started with the quasi-categorical version of the twisted arrow category (and not the complete Segal space version), maybe $\Theta_2$ isn't quite the next step. Instead, let's see what happens if we keep things as close to simplicial sets as we can- say, the complicial route a la Verity. For $n=2$ it's possible to make due with less, and we have Lurie's model of $(\infty,2)$-categories via special types of scaled simplicial sets, i.e. a simplicial set $S$ together with a collection $T \subset S_2$ of thin 2-simplices (which we think of as being 'the invertible 2-morphisms') that contain all the degenerate 2-simplices.

If $\mathcal{C}$ is an $(\infty,2)$-category, we expect the `hom'-functor to be some type of thing like $\mathcal{C}^{op} \times \mathcal{C} \to \mathsf{Cat}_{\infty}$ where the target is the $(\infty,2)$-category of $\infty$-categories. I claim that the appropriate candidate for the twisted arrow category is... the twisted arrow category (again)!

Let me say what I mean more precisely. Given a scaled simplicial set $(S, T)$ form the marked simplicial set $(\mathsf{TwArr}(S), M)$ where $\mathsf{TwArr}(S)_n = S([n]^{op} \star [n])$ and a 1-simplex is marked if and only if the two 2-simplices making up the face of the square are thin. This thing has a projection map to $S^{op} \times S$. I claim that when $S$ is fibrant (i.e. an $(\infty,2)$-category) this is an appropriate sort of fibration which is classified by the lax functor $\mathrm{Map}(-,-): S^{op} \times S \to \mathsf{Cat}_{\infty}$ mentioned above. I'll try to sketch how that goes.

But first, as a sanity check, notice that the fiber above vertices $(X,Y)$ has 0-simplices given by 1-morphisms $X \to Y$ and $1$-simplices given by a little square. Collapsing the edges $(X=X)$ and $(Y=Y)$ leaves you with a little disk with a line through the middle- all arrows oriented left to right. Keeping track of orientations of the 2-simplices tells you that the two 2-arrows glue together to gives a 2-arrow from the bottom to the top (or top to bottom depending on how you drew this.) This 2-morphism need not be thin, so we are seeing the 'category of 2-morphisms'. Not rigorous, but a little reassuring.

1. Lurie proves a version of straightening/unstraightening which produces, for a scaled simplicial set $X$, a Quillen equivalence between a certain model structure on $\left(\mathrm{Set}^{+}_{\Delta}\right)_{/X}$ and a certain model structure on $\mathrm{Fun}(\mathscr{C}^{sc}(X), \mathrm{Set}_{\Delta}^{+})$ (where $\mathscr{C}^{sc}$ is a scaled version of $\mathfrak{C}$). Passing to a scaled version of the nerve, if $X$ is fibrant, produces a map of $(\infty,2)$-categories $X \to \mathsf{Cat}_{\infty}$.
2. The first condition for fibrancy here is that we have an inner fibration on the underlying simplicial sets. I'll skip saying anything about this... you could draw some pictures for $\Lambda^2_1 \subset \Delta^2$ and convince yourself that if the definition of $(\infty,2)$-category says you can 'compose' 2-morphisms in an essentially unique way then you'll be ok. And it does!
3. The next condition for fibrancy is that the marked edges coincide with the locally $p$-cocartesian edges. Intuitively, suppose we have a morphism $(X,Y) \to (X', Y')$ and we've chosen a lift $X \to Y$ of the object $(X,Y)$. Then a morphism covering the given one is the data of a 1-morphism and a 2-morphism from that to the composite $X' \to X \to Y \to Y'$. The universal choice of such is to take "the actual" composite and have the 2-morphism be the "identity". Of course, we're in a weak world, so to do this for real I'd have to tell you more about the definition of an $(\infty,2)$-category.
4. Finally, we need to know that $p$ restricts to a cocartesian fibration along any thin 2-simplex. To check it, you need to know that composing locally cocartesian lifts of each leg gives you a locally cocartesian lift of the 'composite'. Using the characterization of locally cocartesian lifts from before, and the assumption that you're covering a thin 2-simplex, this becomes the statement that 'thin 2-simplices compose to thin 2-simplices' i.e. that 2-equivalences are closed under composition. (Again, really you check some filling condition and so on...).
• Nope. Sorry Dylan. What I care about is the combinatorics of the strict case. Lurie's framework for infinity,n categories suffers from the same problem as ordinary weak n-categories, in that the complexity of the constraints increases without bound. Understanding the $\Theta_2$ case is a way to maybe get an induction started, not a goal in itself Aug 25 '17 at 3:27