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Let $C$ be a category. The twisted arrow category $Tw(C)$ can be refined to a double category $TTw(C)$ by making morphisms on the left "vertical" and morphisms on the right "horizontal".

Question: I'm wondering if this construction, and its relative versions where we splice in a functor, has a name, or has been studied category-theoretically.

Motivation: Variants of the relative version of this construction seem to be useful in algebraic K-theory, allowing one to leverage various powerful tools involving bisimplicial sets (or not -- cf. McCarthy's short, elementary proof of the additivity theorem). This leaves me surprised that I'm not familiar with this construction in a purely category-theoretic context.

As a bisimplicial set: The double nerve of $TTw(C)$ is the bisimplicial set

$NN(TTw(C))([m],[n]) = N(C)([m] \star [n]) = N(C)([m+n+1])$

where $N(C)$ is the nerve of $C$, and $\star$ is the join of simplicial sets. So an $([m],[n])$-cell is a string of composable arrows $c^0_L \to \dots \to c^m_L \to c^0_R \to \dots \to c^n_R$. The diagaonal recovers the usual twisted arrow category: $d NN(TTw(C)) = N(Tw(C))$.

More generally, if $F: D \to C$ and $G: E \to C$ are functors, we may define $TTw(F,G)([m],[n])$ to consist of a string $d_0 \to \dots \to d_m$ in $D$, a string $e_0,\dots, e_n$ in $E$, and a connecting map $F(d_m) \to G(e_0)$ in $C$.

More generally still, if $M: D^{op} \times E \to \mathsf{Set}$ is a bimodule/profunctor/correspondence from $D$ to $E$, we can form a similar double category $TTw(M)$ where $TTw(M)([m],[n])$ consists of an $[m]$-string $\vec d$ in $D$, an $[n]$-string $\vec e$ in $E$, and some $x \in M(d_n,e_0)$.

As a double category: $TTw(C)$ may be described as follows (I'll stick to the non-relative version for simplicicty):

  • An object in $TTw(C)$ is a morphism $c_L \overset f \to c_R$ in $C$.

  • A vertical morphism from $c_L \overset f \to c_R$ to $c_L' \overset {f'} \to c_R'$ exists only if $c_R = c_R'$ and consists of a morphism $c_L' \to c_L$ (note the direction) such that the obvious diagram commutes.

  • A horizontal morphism from $c_L \overset f \to c_R$ to $c_L' \overset {f'} \to c_R'$ exists only if $c_L = c_L'$ and consists of a morphism $c_R \to c_R'$ (note the direction) such that the obvious diagram commutes.

  • A square in $TTw(C)$ is a commutative diagram:

$\require{AMScd} \begin{CD} c_L @>{f}>> c_R \\ @A{\gamma_L}AA @VV{\gamma_R}V\\ c_L' @>{f'}>> c_R' \end{CD}$

This is interpreted as the following double-categorical square:

$\require{AMScd} \begin{CD} f @>{\gamma_R}>> \gamma_R f \\ @V{\gamma_L}VV @VV{\gamma_L}V \\ f \gamma_L @>{\gamma_R}>> f' \end{CD}$

Note: $TTw(C)$ is what BOORS call a stable double category: a square $\begin{CD} A @>>> B \\ @VVV @VVV \\ C @>>> D \end{CD}$ may be recovered from knowing either the following 3 objects, horizontal leg, and vertical leg:

$\begin{CD} A @>>> B \\ @. @VVV \\ @. D \end{CD}$

or the following 3 objects, vertical leg, and horizontal leg:

$\begin{CD} A @. \\ @VVV @. \\ C @>>> D \end{CD}$

In fact, on reflection it seems pretty clear that this construction shows that a stable double category is the same thing as a pair of categories $D,E$ with a bimodule $M: D^{op} \times E \to \mathsf{Set}$ between them.

In the BOORS framework, the operation taking a stable double category to its diagonal "category" is essentially the $S_\bullet$ construction, and it recovers $Tw(C)$ from $TTw(C)$.

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    $\begingroup$ This is not exactly an answer, but I used this notion to study $\infty$-categories themselves. Namely in Section 5.6 here: mathematik.uni-regensburg.de/cisinski/CatLR.pdf Section 5.5 also contains material on bisimplicial sets which is likely to help proving what you describe in your post but for $\infty$-categories. $\endgroup$ – Denis-Charles Cisinski Mar 15 '18 at 21:38
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    $\begingroup$ As a bisimplicial set, it is also called the total décalage of (the nerve of) $C$. $\endgroup$ – Thomas Poguntke Mar 16 '18 at 12:18
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    $\begingroup$ Note that the construction of a double category from a bimodule $D^{\mathrm{op}}\times E \to\mathrm{Set}$ can be regarded as a double-categorical Grothendieck construction of a double functor $H D^{\mathrm{op}}\times V E \to Q \mathrm{Set}$, where $H,V,Q$ mean regarding a category as a double category horizontally, vertically, or using quintets (and commutative squares). $\endgroup$ – Mike Shulman Mar 18 '18 at 10:10

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