# Join/slice adjunction and lifting problems

Let $$C \in sSet$$ be a quasicategory. $$*$$ denote the join of simplicial sets. Then consider $$- * X \colon sSet \to sSet_{X/}$$, where $$sSet_{X/}$$ is the slice category of simplicial set under $$X$$, with objects $$X \to Y$$.

For $$p \colon X \to Y$$, $$X_{/p}$$ is $$p$$ evaluated at right adjoint of $$- * X$$.

$$C_{/x}$$ is $$x \colon \triangle^0 \to C$$ evaluated at right adjoint of $$-*\triangle^0$$.

I'm trying to make sense of the bijective correspondence as shown below.

I get that $$\triangle^n \to C_{/x}$$ correspond to $$\triangle^n * \triangle^0 \to C$$, but I have no idea how other maps correspond.

Any help would be appreciated, thanks!

(This is adapted from Chapter 3 of "The Theory of Quasicategories and Its Applications Vol. 2" by Andre Joyal. Warning: The notation is different.)

## Notation and Background

Fix a simplicial set $$T \in \textsf{sSet}$$. Let $$T \downarrow \textsf{sSet}$$ denote the category whose objects are simplicial maps $$T \to X$$ and whose morphisms are commuting triangles.

We have a functor $$\left(-\right) \ast T \colon \textsf{sSet} \to T \downarrow \textsf{sSet}$$ that maps a simplicial set $$X$$ to the object $$T = \emptyset \ast T \to X \ast T$$ in the slice category. This functor admits a right adjoint $$T \downarrow \left(-\right) \colon T \downarrow \textsf{sSet} \to \textsf{sSet}$$

Given an object $$t \colon T \to X$$ in $$T \downarrow \textsf{sSet}$$, the $$n$$-simplices of $$T \downarrow t$$ are given by: $$\left(T \downarrow t\right)_{n} = \textsf{sSet}\left(\Delta^{n},T \downarrow t\right) = \left(T \downarrow \textsf{sSet}\right)\left(T \to \Delta^{n} \ast T, t \colon T \to X\right)$$. When the context is clear, we will denote objects $$t \colon T \to X$$ in the slice category $$T \downarrow \textsf{sSet}$$ by their codomain $$X$$. For instance, we write $$T \downarrow X$$ instead of $$T \downarrow t$$.

Now, consider a simplicial map $$S \xrightarrow{j} T$$. We have an induced natural transformation $$j \downarrow \left(-\right) \colon T \downarrow \left(-\right) \to S \downarrow \left(-\right)$$ On an object $$t \colon T \to X$$ in $$T \downarrow \textsf{sSet}$$, this is given on $$n$$-simplices by: $$\left(j \downarrow X\right)_{n} \colon \left(T \downarrow X\right)_{n} \to \left(S \downarrow X\right)_{n}$$ $$\left(\Delta^{n} \ast T \xrightarrow{x} X\right) \mapsto \left( \Delta^{n} \ast S \xrightarrow{\Delta^{n} \ast j} \Delta^{n} \ast T \xrightarrow{x} X\right)$$

Given simplicial maps $$S \xrightarrow{j} T \xrightarrow{t} X \xrightarrow{f} Y$$, the naturality square $$\require{AMScd}$$ $$\begin{CD} T \downarrow X @>j \downarrow X>> S \downarrow X \\ @V T \downarrow f VV @VV S \downarrow f V \\ T\downarrow Y @>>j \downarrow j> S \downarrow Y \end{CD}$$ induces the pulbback slice $$j \vartriangleright f \colon T \downarrow X \to \left(T \downarrow Y\right) \times_{S \downarrow Y} \left(S \downarrow X\right)$$

## Lemma

Given simplicial maps $$i \colon A \to B$$, $$j \colon S \to T$$, $$t \colon T \to X$$, and $$f \colon X \to Y$$, we have a 1-1 correspondence of lifting problems: $$\require{AMScd}$$ $$\begin{CD} \left(A \ast T\right) \cup_{A \ast S} \left(B \ast S\right) @> >> X \\ @V i \hat{\ast} j VV @VV f V \\ B \ast T @>> > Y \end{CD} \longleftrightarrow \begin{CD} A @> >> T \downarrow X \\ @V i VV @VV j \vartriangleright f V \\ B @>> > \left(T \downarrow Y\right) \times_{S \downarrow Y} \left(S \downarrow X\right) \end{CD}$$ Moreover, there is a 1-1 correspondence between their solutions. In particular, $$\left(i \hat{\ast} j\right) \pitchfork f \Leftrightarrow i \pitchfork \left(j \vartriangleright f\right)$$.

(The correspondence is given by repeatedly using the adjunction between join and slice. This is Lemma 3.14 in Joyal's "The Theory of Quasicategories and Its Applications Vol. 2".)

$$C_{/x}$$ in your question corresponds to $$\Delta^{0} \downarrow C$$ in my notation. That is, the functor $$\Delta^{0} \downarrow \left(-\right)$$ applied to the object $$x \colon \Delta^{0} \to C$$.
Now, in the above lemma, take $$i \colon A \to B$$ to be the horn inclusion $$v^{n}_{j} \colon \Lambda^{n}_{j} \hookrightarrow \Delta^{n}$$, take $$j \colon S \to T$$ to be the inclusion $$\emptyset \hookrightarrow \Delta^{0}$$, $$t \colon T \to X$$ to be the 0-cell $$x \colon \Delta^{0} \to C$$, and $$f \colon X \to Y$$ to be unique map $$C \xrightarrow{!} \Delta^{0}$$.
Note that $$\left(\emptyset \hookrightarrow \Delta^{0}\right) \vartriangleright \left(C \xrightarrow{!} \Delta^{0}\right)$$ is precisely the map $$\pi \colon C_{/x} \to C$$ in your question.