I am searching to show a quite technical result and I am wondering the following. Suppose $p: C \to D$ is a functor of infinity categories.

  1. Take a cell $\Delta^2 \to C$, and suppose that $\Delta^{\{0,1\}}, \Delta^{\{0,2\}}$ are p-cocartesian. Is it true that $\Delta^{\{1,2\}}$ is p-cocartesian? Some low dimensional examples seems to give a positive answer, but I can't find a general argument not-by-explicit-computation.

  2. Suppose $f:x \to y \in D$ is an arrow, and fix liftings $X,Y$ of the ending points. Is it true that cocartesian liftings of $f$ as a subsimplicial set of $Map(X,Y)$ are a contractible sSet?

We would have the following corollaries:

  1. Suppose $\sigma: \Delta^2 \to D$ is a cell and $j: \Lambda^2_1 \to C$ is a horn over $\sigma$ made of cocartesian edges. Then the infinity category of fillings of $j$ over $\sigma$ is contractible.

Proof: Indeed, every filling has a third edge which must be cocartesian by (1). Moreover, it lies over a fixed edge in $\sigma$, and has fixed ending points. Thus, by (2), the sSet of such fillings is contractible.

  1. Take $H:X \to Y$ to be a cocartesian lift of $h: x \to y$ in $D$. Suppose $h$ is strictly final in $D_{x/}$. Then $(C_{X/})^{cocar}$ is weakly contractible.

Proof: Take a $G:X \to Z$ in $(C_{X/})^{cocar}$ be a cocartesian lift of its image $g:x \to z$. By hypothesis, there exist a unique $f:z \to y$ from $g$ to $h$. Thus the set of maps from $G$ to $H$ lies over a fixed $f$, and must be cocartesian by (1). It follows that it is contractible. But then $H$ is final in $(C_{X/})^{cocar}$, so the latter is weakly contractible.

Thank you! Best, Andrea

  1. If $p$ is an inner fibration then the answer is yes (HTT. Lurie doesn't even define p-cocartesian unless $p$ is an inner fibration, but it's not hard to redefine the notion slightly more generally so as to be homotopy invariant (just replace the fiber product in the definition of p-cocartesian with a homotopy fiber product in the Joyal model structure, say).
  2. No. Take D to be a point and take X=Y. Then every automorphism of X is a cocartesian lift of the identity, but often the space of automorphisms of X will not be contractible. If instead you fix X but not Y then the answer is yes: for fixed arrow downstairs and fixed lift of the source, there is a contractible space of cocartesian lifts. (That is immediate from the definition of cocartesian.)
  • $\begingroup$ Thank you! What about the two corollaries? Are they still true? $\endgroup$ – Andrea Marino May 29 '19 at 9:06

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