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2.3.5 Twisted Arrows and the Nerve of $\operatorname{Corr}(\operatorname{\mathcal{C}})^{\operatorname{c}}$

Let $\operatorname{\mathcal{E}}$ be a category which admits fiber products and let $\operatorname{Corr}(\operatorname{\mathcal{E}})$ denote the $2$-category of correspondences in $\operatorname{\mathcal{E}}$ (Example 2.2.2.1). Our goal in this section is to give an explicit description of the Duskin nerve of the conjugate $2$-category $\operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}}$ (Corollary 2.3.5.8 and Remark 2.3.5.9). We will obtain this description by formulating a universal property of $\operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}}$ as an object of the category $\operatorname{2Cat}_{\operatorname{ULax}}$. First, we need a brief digression.

Construction 2.3.5.1 (The Twisted Arrow Category). Let $\operatorname{\mathcal{C}}$ be a category. We define a new category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as follows:

  • An object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is a morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$.

  • Let $f: C \rightarrow D$ and $f': C' \rightarrow D'$ be objects of $\operatorname{Tw}(\operatorname{\mathcal{C}})$. A morphism from $f$ to $f'$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is a pair of morphisms $u: C \rightarrow C'$, $v: D' \rightarrow D$ in $\operatorname{\mathcal{C}}$ satisfying $f = v \circ f' \circ u$, so that we have a commutative diagram

    \[ \xymatrix { C \ar [r]^{f} \ar [d]^{u} & D \\ C' \ar [r]^{f'} & D'. \ar [u]_{v} } \]
  • Let $f: C \rightarrow D$, $f': C' \rightarrow D'$, and $f'': C'' \rightarrow D''$ be objects of $\operatorname{Tw}(\operatorname{\mathcal{C}})$. If $(u,v)$ is a morphism from $f$ to $f'$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ and $(u',v')$ is a morphism from $f'$ to $f''$ in $\operatorname{\mathcal{C}}$, then the composition $(u',v') \circ (u, v)$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is the pair $(u' \circ u, v \circ v')$.

We will refer to $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as the twisted arrow category of $\operatorname{\mathcal{C}}$.

Remark 2.3.5.2. Let $[1] = \{ 0 < 1 \} $ denote a linearly ordered set with two elements. For any category $\operatorname{\mathcal{C}}$, we can identify morphisms of $\operatorname{\mathcal{C}}$ with functors $F: [1] \rightarrow \operatorname{\mathcal{C}}$. The collection of such functors can be organized into a category $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$, which we refer to as the arrow category of $\operatorname{\mathcal{C}}$. The arrow category $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ has the same objects as the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. However, the morphisms are different: if $f: C \rightarrow D$ and $f': C' \rightarrow D'$ are morphisms of $\operatorname{\mathcal{C}}$, then morphisms from $f$ to $f'$ in $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ can be identified with commutative diagrams

\[ \xymatrix { C \ar [r]^{f} \ar [d] & D \ar [d] \\ C' \ar [r]^{f'} & D'. } \]

Remark 2.3.5.3. Let $\operatorname{\mathcal{C}}$ be a category and let $\operatorname{Tw}(\operatorname{\mathcal{C}})$ denote its twisted arrow category. There is an evident forgetful functor $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}^{\operatorname{op}}$, given on objects by the construction $(f: C \rightarrow D) \mapsto (C,D)$.

Example 2.3.5.4. Let $Q$ be a partially ordered set, which we regard as a category. Then the twisted arrow category $\operatorname{Tw}(Q)$ can be identified (via the forgetful functor of Remark 2.3.5.3) with the partially ordered set

\[ \{ (p,q) \in Q \times Q^{\operatorname{op}}: p \leq q \} \subseteq Q \times Q^{\operatorname{op}}. \]

Remark 2.3.5.5 (Functoriality). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of categories. Then $F$ induces a functor of twisted arrow categories $\operatorname{Tw}(F): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}})$. This construction is compatible with composition, and can therefore be regarded as a functor $\operatorname{Tw}: \operatorname{Cat}\rightarrow \operatorname{Cat}$ from the category of (small) categories to itself.

Construction 2.3.5.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be categories, where $\operatorname{\mathcal{E}}$ admits fiber products, and let $F: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be a functor. We define a strictly unitary lax functor $F^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}}$ as follows:

  • For each $C \in \operatorname{\mathcal{C}}$, we define $F^{+}(C) = F( \operatorname{id}_{C} )$; here we regard the identity morphism $\operatorname{id}_ C: C \rightarrow C$ as an object of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$.

  • For each morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, we define $F^{+}(f)$ to be the $1$-morphism of $\operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}}$ given by the correspondence

    \[ F( \operatorname{id}_ C ) \xleftarrow { F(\operatorname{id}_ C, f) } F(f) \xrightarrow { F( f, \operatorname{id}_ D) } F( \operatorname{id}_ D ); \]

    this determines the values of $F^{+}$ on $2$-morphisms, since every $2$-morphism in $\operatorname{\mathcal{C}}$ is an identity $2$-morphism.

  • For every pair of composable morphisms $C \xrightarrow {f} D \xrightarrow {g} E$, the composition constraint $\mu _{g,f}: F^{+}(g) \circ F^{+}(f) \Rightarrow F^{+}(g \circ f)$ is the $2$-morphism of $\operatorname{Corr}(\operatorname{\mathcal{C}})^{\operatorname{c}}$ corresponding to the map $F(g \circ f) \rightarrow F(f) \times _{ F(\operatorname{id}_ Y) } F(g)$ classified by the commutative diagram

    \[ \xymatrix { & F(g \circ f) \ar [dl]_{ F(\operatorname{id}_ C, g)} \ar [dr]^{ F( f, \operatorname{id}_ E) } & \\ F(f) \ar [dr]_{ F(f, \operatorname{id}_ D) } & & F(g) \ar [dl]^{ F( \operatorname{id}_ D, g) } \\ & F( \operatorname{id}_ D) & } \]

    in the category $\operatorname{\mathcal{E}}$.

Theorem 2.3.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be categories where $\operatorname{\mathcal{E}}$ admits fiber products. Then Construction 2.3.5.6 induces a bijection of sets

\[ \{ \textnormal{Functors $F: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$} \} \xrightarrow {\sim } \{ \textnormal{Strictly Unitary Lax Functors $F^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}}$} \} . \]

Corollary 2.3.5.8. Let $\operatorname{\mathcal{E}}$ be a category which admits fiber products and let $\operatorname{Corr}(\operatorname{\mathcal{E}})$ denote the $2$-category of correspondences in $\operatorname{\mathcal{E}}$ (Example 2.2.2.1). Then the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}} )$ is given concretely by the formula

\[ ( [n] \in \operatorname{{\bf \Delta }}) \mapsto \{ \textnormal{Functors $\operatorname{Tw}([n]) \rightarrow \operatorname{\mathcal{E}}$} \} . \]

Remark 2.3.5.9. Let $\operatorname{\mathcal{E}}$ be a category which admits fiber products. Combining Corollary 2.3.5.8 with Example 2.3.5.4, we see that $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}} )$ can be identified with diagrams

\[ \{ (i,j) \in [n] \times [n]^{\operatorname{op}}: i \leq j \} \rightarrow \operatorname{\mathcal{E}}, \]

which we can represent graphically as

\[ \xymatrix { & & & X_{0,n} \ar [dl] \ar [dr] & & & \\ & & \cdots \ar [dl] \ar [dr] & & \cdots \ar [dl] \ar [dr] & & \\ & X_{0,1} \ar [dl] \ar [dr] & & \cdots \ar [dl]\ar [dr] & & X_{n-1,n} \ar [dl] \ar [dr] & \\ X_{0,0} & & X_{1,1} & \cdots & X_{n-1,n-1} & & X_{n,n}. } \]

In ยง, we will use this description to extend the definition of $\operatorname{Corr}(\operatorname{\mathcal{E}})$ to the case where $\operatorname{\mathcal{E}}$ is an $\infty $-category.

Example 2.3.5.10. Let $\operatorname{\mathcal{E}}$ be a category which admits fiber products. Then $2$-simplices $\sigma $ of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}} )$ can be identified with commutative diagrams

\[ \xymatrix { & & X_{0,2} \ar [dl] \ar [dr] & & \\ & X_{0,1} \ar [dl] \ar [dr] & & X_{1,2} \ar [dl] \ar [dr] & \\ X_{0,0} & & X_{1,1} & & X_{2,2} } \]

in the category $\operatorname{\mathcal{E}}$. Such a diagram corresponds to a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}} )$ (in the sense of Definition 2.3.2.3) if and only if the square appearing in the diagram is a pullback: that is, it induces an isomorphism $X_{0,2} \rightarrow X_{0,1} \times _{ X_{1,1} } X_{1,2}$.

Proof of Theorem 2.3.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be categories, where $\operatorname{\mathcal{E}}$ admits fiber products, and let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}}$ be a strictly unitary lax functor of $2$-categories. For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, we can identify $G(f)$ with a correspondence from $G(C)$ to $G(D)$ in the category $\operatorname{\mathcal{E}}$, given by a diagram we will denote by $G(C) \xleftarrow { u(f) } M(f) \xrightarrow { v(f) } G(D)$. Our assumption that $G$ is strictly unitary guarantees the following:

$(\ast )$

For each object $C \in \operatorname{\mathcal{C}}$, the object $M( \operatorname{id}_ C )$ is equal to $G(C)$, and the maps $u(\operatorname{id}_ C): M( \operatorname{id}_ C) \rightarrow G(C)$ and $v( \operatorname{id}_ C): M( \operatorname{id}_ C) \rightarrow G(C)$ are the identity morphisms from $G(C)$ to itself in the category $\operatorname{\mathcal{E}}$.

For every pair of composable morphisms $C \xrightarrow {f} D \xrightarrow {g} E$, the composition constraint $\mu _{g,f}$ for the lax functor $G$ can be identified with a morphism from $M(g \circ f)$ to the fiber product $M(f) \times _{ G(D)} M(g)$ in the category $\operatorname{\mathcal{E}}$, or equivalently with a pair of morphisms

\[ p(g,f): M(g \circ f) \rightarrow M(f) \quad \quad q(g,f): M(g \circ f) \rightarrow M(g) \]

satisfying $v(f) \circ p(g,f) = u(g) \circ q(g,f)$. The axioms for a lax functor (Definition 2.2.4.4) then translate to the following additional conditions:

$(a)$

For every $1$-morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, $p( \operatorname{id}_ D, f)$ is the identity morphism from $M(f)$ to itself.

$(b)$

For every $1$-morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, $q( f, \operatorname{id}_ C )$ is the identity morphism from $M(f)$ to itself.

$(c)$

For every composable triple of $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ in the category $\operatorname{\mathcal{C}}$, we have

\[ p(h \circ g, f) = p(g,f) \circ p( h, g \circ f) \quad \quad q(h, g \circ f ) = q(h,g) \circ q(h \circ g, f) \]
\[ q(g,f) \circ p( h, g \circ f) = p( h, g) \circ q(h \circ g, f). \]

We wish to show that there exists a unique functor of ordinary categories $F: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ such that $G = F^{+}$, where $F^{+}$ is the lax functor associated to $F$ by Construction 2.3.5.6. For this condition to be satisfied, the functor $F$ must have the following properties:

$(0)$

For each object $C \in \operatorname{\mathcal{C}}$, we have $F( \operatorname{id}_ C ) = G(C)$ (this guarantees that $G$ and $F^{+}$ coincide on objects).

$(1)$

For each morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$ (regarded as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$), we have $F(f) = M(f)$, and the morphisms $u(f)$ and $v(f)$ are given by $F( \operatorname{id}_ C, f)$ and $F(f, \operatorname{id}_ D)$, respectively (this guarantees that $G$ and $F^{+}$ coincide on $1$-morphisms, and therefore also on $2$-morphisms).

$(2)$

For every pair of composable morphisms $C \xrightarrow {f} D \xrightarrow {g} E$, the morphisms $p(g,f): M(g \circ f) \rightarrow M(f)$ and $q(g,f): M(g \circ f) \rightarrow M(g)$ are given by $F(\operatorname{id}_ C,g): F(g \circ f) \rightarrow F(f)$ and $F(f, \operatorname{id}_ E): F(g \circ f) \rightarrow F(g)$, respectively (this guarantees that the composition constraints on $G$ and $F^{+}$ coincide).

Note that the value of $F$ on each object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is determined by condition $(1)$. Moreover, if $(u,v)$ is a morphism from $f: C \rightarrow D$ to $f': C' \rightarrow D'$ in the category $\operatorname{Tw}(\operatorname{\mathcal{C}})$, then condition $(2)$ guarantees that $F(u,v)$ must be equal to the composition

\[ F(f) = M(v \circ f' \circ u) \xrightarrow { p(v, f' \circ u)} M(f' \circ u) \xrightarrow {q(f',u)} M(f') = F(f'). \]

This proves the uniqueness of the functor $F$.

To prove existence, we define $F$ on objects $f$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ by the formula $F(f) = M(f)$, and on morphisms $(u,v): f \rightarrow f'$ by the formula $F(u,v) = q(f',u) \circ p(v, f' \circ u)$. For any morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, we can use $(a)$ and $(b)$ to compute

\[ F(\operatorname{id}_ C, \operatorname{id}_ D) = q( f, \operatorname{id}_ C) \circ p( \operatorname{id}_ D, f \circ \operatorname{id}_ C) = \operatorname{id}_{M(f)} \circ \operatorname{id}_{ M(f)} = \operatorname{id}_{ F(f) }, \]

so that $F$ carries identity morphisms in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to identity morphisms in $\operatorname{\mathcal{E}}$. To complete the proof that $F$ is a functor, we note that for every pair of composable morphisms

\[ (f: C \rightarrow D) \xrightarrow { (u,v)} (f': C' \rightarrow D') \xrightarrow { (u',v')} (f'': C'' \rightarrow D'') \]

in $\operatorname{Tw}(\operatorname{\mathcal{C}})$, the identities given in $(c)$ allow us to compute

\begin{eqnarray*} F(u', v') \circ F(u,v) & = & q( f'', u') \circ p(v', f'' u') \circ q(v' f'' u', u) \circ p( v, v' f'' u' u) \\ & = & q(f'', u') \circ q(f'' u', u) \circ p( v', f'' u' u ) \circ p(v, v' f'' u' u) \\ & = & q( f'', u' u ) \circ p( v' v, f'' u' u) \\ & = & F(u' \circ u, v \circ v' ). \end{eqnarray*}

We now complete the proof by showing that the functor $F$ satisfies conditions $(0)$, $(1)$, and $(2)$. Condition $(0)$ is an immediate consequence of $(\ast )$. To prove $(2)$, we note that for any pair of composable morphisms $C \xrightarrow {f} D \xrightarrow {g} E$, identities $(a)$ and $(b)$ yield equalities

\[ F( \operatorname{id}_ C, g) = q(f,\operatorname{id}_ C) \circ p( g, f \circ \operatorname{id}_ C) = p(g,f) \quad \quad F(f, \operatorname{id}_ E) = q(g,f) \circ p( \operatorname{id}_ E, g \circ f) = q(g,f). \]

For $(1)$, we note that if $f: C \rightarrow D$ is a morphism in $\operatorname{\mathcal{C}}$, then we have

\begin{eqnarray*} F(\operatorname{id}_ C, f) & = & q( \operatorname{id}_ C, \operatorname{id}_ C) \circ p( f, \operatorname{id}_ C \circ \operatorname{id}_ C) \\ & = & \operatorname{id}_{G(C)} \circ p(f, \operatorname{id}_ C ) \\ & = & v( \operatorname{id}_ C ) \circ p( f, \operatorname{id}_ C) \\ & = & u(f) \circ q(f, \operatorname{id}_ C) \\ & = & u(f) \end{eqnarray*}

and a similar calculation yields $F(f, \operatorname{id}_ D) = v(f)$. $\square$