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As so often, I'm failing to construct a map in $\infty$-category theory that is easily constructed in $1$-category theory.

Let $\mathscr{C}$ be an $\infty$-category and let $F: \mathscr{C} \to \mathsf{Cat}_{\infty}$ such that every $Fc$ has an initial object $\emptyset_{Fc}$ for $c \in \mathscr{C}$. Furthermore, suppose that $Ff$ preserves initial objects for maps $f$ in $\mathscr{C}$. How do I construct (prove the existence of) a natural transformation $\mathsf{const}_* \Rightarrow F$ which picks out $\emptyset_{Fc}$ levelwise?

This is the datum of a simplicial map $\mathscr{C} \times [1] \to \mathsf{Cat}_{\infty}$ but there are infinitely many coherences, so I couldn't manage to construct such a natural transformation. I would also be happy with a reference.

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    $\begingroup$ Do we know that $Ff$ preserves initial objects for any $f\colon c\to d$ in $\mathscr{C}$? Otherwise such a natural transformation does not exist. $\endgroup$ Commented May 19 at 13:37
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    $\begingroup$ @DaniëlApol You're right! I added this condition. $\endgroup$
    – Qi Zhu
    Commented May 19 at 14:01

1 Answer 1

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Let $\mathscr{C}^\triangleleft:\simeq[0]\star\mathscr{C}$ be the $\infty$-category obtained from $\mathscr{C}$ by adjoining an initial object, and write $-\infty$ for this initial object. It suffices to construct a functor $F^\triangleleft\colon\mathscr{C}^\triangleleft\to\mathsf{Cat}_\infty$ such that:

  1. $F(-\infty)\simeq*$.
  2. $F^\triangleleft|_{\mathscr{C}}\simeq F$.
  3. For every $c\in\mathscr{C}$, the unique map $i\colon\varnothing\to c$ in $\mathscr{C}^\triangleleft$ is sent by $F^\triangleleft$ to $\varnothing_{Fc}\colon *\to Fc$.

Let $p\colon\mathscr{E}\to\mathscr{C}$ be the cocartesian fibration associated to $F$. We claim that $p^\triangleleft\colon\mathscr{E}^\triangleleft\to\mathscr{C}^\triangleleft$ is a cocartesian fibration, and that the associated functor $F^\triangleleft\colon\mathscr{C}^\triangleleft\to\mathsf{Cat}_\infty$ is the desired one. In fact, it is already clear that point 1. above is satisfied. This means that we have to show the following:

a) $p^\triangleleft\colon\mathscr{E}^\triangleleft\to\mathscr{C}^\triangleleft$ is a cocartesian fibration.

b) The commutative square $$ \require{AMScd} \begin{CD} \mathscr{E} @>>> \mathscr{E}^\triangleleft\\ @V{p}VV @VV{p^\triangleleft}V\\ \mathscr{C} @>>> \mathscr{C}^\triangleleft \end{CD} $$ is a pullback square in $\mathsf{Cat}_\infty$. Here, the maps $\mathscr{C}\to\mathscr{C}^\triangleleft$ and $\mathscr{E}\to\mathscr{E}^\triangleleft$ are the canonical fully faithful functors of this form.

c) Writing $\varnothing$ for the initial object of $\mathscr{E}^\triangleleft$, the unique morphism $i\colon\varnothing\to\varnothing_{Fc}$ in $\mathscr{E}^\triangleleft$ is $p^\triangleleft$-cocartesian.

Indeed, points b) and c) imply points 2. and 3. under straightening-unstraightening, and we of course need point a) to even be allowed to use straightening-unstraightening.

We will start with proving b), as it is needed in the proofs of c) and a). If we write $\mathscr{P}:\simeq\mathscr{C}\times_{\mathscr{C}^\triangleleft}\mathscr{E}^\triangleleft$, then the canonical map $\mathscr{E}\to\mathscr{P}$ is easily seen to be fully faithful and essentially surjective.

Now we prove c), as it is needed in the proof of a). We need to show that the commutative diagram $$ \require{AMScd} \begin{CD} \mathscr{E}^\triangleleft_{\varnothing_{Fc}/} @>{i^*}>>\mathscr{E}^\triangleleft_{\varnothing/}\\ @VVV @VVV\\ \mathscr{C}^\triangleleft_{c/} @>>> \mathscr{C}^\triangleleft_{-\infty/} \end{CD} $$ is a pullback square in $\mathsf{Cat}_\infty$. It is not hard to see that the canonical map $\mathscr{C}^\triangleleft_{c/}\to\mathscr{C}_{c/}$ is an equivalence (as $c\in\mathscr{C}$). Moreover, the forgetful map $\mathscr{C}^\triangleleft_{-\infty/}\to\mathscr{C}^\triangleleft$ is an equivalence since $-\infty$ is initial in $\mathscr{C}^\triangleleft$. Using similar statements for $\mathscr{E}$ instead of $\mathscr{C}$, we see that it suffices to show that the square $$ \require{AMScd} \begin{CD} \mathscr{E}_{\varnothing_{Fc}/} @>>> \mathscr{E}^\triangleleft\\ @VVV @VVV\\ \mathscr{C}_{c/} @>>> \mathscr{C}^\triangleleft \end{CD} $$ is a pullback square. In light of statement b), it suffices to show that the square $$ \require{AMScd} \begin{CD} \mathscr{E}_{\varnothing_{Fc}/} @>{\mathrm{forget}}>> \mathscr{E}\\ @VVV @VVV\\ \mathscr{C}_{c/} @>>{\mathrm{forget}}> \mathscr{C} \end{CD} $$ is a pullback square. Write $\mathscr{Q}$ for the pullback of the cospan appearing above. Since the canonical map $\mathscr{E}_{\varnothing_{Fc}/}\to\mathscr{Q}$ is a map of cocartesian fibrations over $\mathscr{C}_{c/}$ by construction, we can check that it is an equivalence on fibers of $\mathscr{C}_{c/}$. Given $f\colon c\to d$, considered as an object of the latter, a small bit of pullback pasting shows that we need to show that the map $$ (\mathscr{E}_{\varnothing_{Fc}/})_f\to\mathscr{E}_d $$ induced by the forgetful functor $\mathscr{E}_{\varnothing_{Fc}/}\to\mathscr{E}$ is fully faithful and essentially surjective. Let us first show the latter. Given an object $e\in\mathscr{E}$ living over $d\in\mathscr{C}$, we use the fact that there is a $p$-cocartesian morphism $\varnothing_{Fc}\to\varnothing_{Fd}$ in $\mathscr{E}$ lying over $f$ (here we use the fact that $Ff\colon Fc\to Fd$ preserves initial objects, although we do not use yet the fact that this morphism in $\mathscr{E}$ is actually $p$-cocartesian). Since $\varnothing_{Fd}$ is initial in $\mathscr{E}_d$, there is a morphism $\varnothing_{Fd}\to e$ in $\mathscr{E}$ living over $\mathrm{id}_d$. The composite $\varnothing_{Fc}\to\varnothing_{Fd}\to d$ in $\mathscr{E}$ therefore lives over $f$, and is an object of $(\mathscr{E}_{\varnothing_{Fc}/})_f$ that is sent to $e\in\mathscr{E}_d$ under our map above.

In order to see that our map is fully faithful, we pick two maps $\varphi\colon\varnothing_{Fc}\to e$ and $\psi\colon\varnothing_{Fc}\to e'$ in $\mathscr{E}$ lying over $f$. We must show that the induced morphism $$ (\mathscr{E}_{\varnothing_{Fc}/})_f(\varphi,\psi)\to\mathscr{E}_d(e,e') $$ is an equivalence of anima. For this, we consider the commutative diagram enter image description here The top and bottom squares of the cube are pullbacks, and we claim that the map $\mathscr{E}(\varnothing_{Fc},e')\to\mathscr{C}(c,d)$ is an equivalence. Granting this, the front square of the cube is also a pullback, and hence the back square as well. The fact that the backside outer rectangle is a pullback implies that $(\mathscr{E}_{\varnothing_{Fc}/})_f(\varphi,\psi)\to\mathscr{E}_d(e,e')$ is an equivalence. It remains to check that $\mathscr{E}(\varnothing_{Fc},e')\to\mathscr{C}(c,d)$ is an equivalence. Now we will actually use that we have a $p$-cocartesian map $g\colon\varnothing_{Fc}\to\varnothing_{Fd}$ in $\mathscr{E}$ living over $f$ (to reiterate, this follows from the fact that $Ff\colon Fc\to Fd$ preserves initial objects). This namely implies that we have a pullback square $$ \require{AMScd} \begin{CD} \mathscr{E}_d(\varnothing_{Fd},e') @>{g^*}>>\mathscr{E}(\varnothing_{Fc},e')\\ @VVV @VVV\\ * @>>{f}> \mathscr{C}(c,d) \end{CD} $$ However, since $\varnothing_{Fd}$ is initial in $\mathscr{E}_d$, this shows that all fibers of the map $\mathscr{E}(\varnothing_{Fc},e')\to\mathscr{C}(c,d)$ of anima are contractible, so that this map is an equivalence. This means we have finally proven statement c).

We end with showing a). In light of part c), we only need to show the following: if $h\colon e\to e'$ is a $p$-cocartesian morphism living over $f\colon c\to d$ in $\mathscr{C}$, then $h$ is also $p^\triangleleft$-cocartesian. However, as we remarked earlier in the proof of part c), the commutative square $$ \require{AMScd} \begin{CD} \mathscr{E}^\triangleleft_{e'/} @>{h^*}>> \mathscr{E}^\triangleleft_{e/}\\ @V{p^\triangleleft}VV @VV{p^\triangleleft}V\\ \mathscr{C}^\triangleleft_{c/} @>>{f^*}> \mathscr{C}^\triangleleft_{d/} \end{CD} $$ of which we have to prove that it is a pullback is equivalent to the commutative square $$ \require{AMScd} \begin{CD} \mathscr{E}_{e'/} @>{h^*}>> \mathscr{E}_{e/}\\ @V{p}VV @VV{p}V\\ \mathscr{C}_{c/} @>>{f^*}> \mathscr{C}_{d/} \end{CD} $$ which is a pullback by assumption that $h$ is $p$-cocartesian. This proves part a), and this means we are done.

As you see, the $\infty$-categorical construction (of something that exists trivially in $1$-category land) only required a tiny adjustment of the $1$-categorical argument ;-).

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