My question: is it true in general for a small category $\mathcal{C}$ that the canonical functor
\begin{aligned}
\operatorname{colim}_{c\in \mathcal{C}}\mathcal{C}/x \to \mathcal{C}
\end{aligned}
is an equivalence of categories? If so, is there a canonical way of seeing this without explicitly constructing an inverse?
Let me explain the statement in more detail. For an object $x$ in $\mathcal{C}$, let $C/x$ denote the over-category of $\mathcal{C}$ over $x$, with objects given by morphisms $f: y \to x$ in $\mathcal{C}$ and morphisms from $f: y \to x$ to $f': y'\to x$ given by morphisms $g: y \to y'$ in $\mathcal{C}$ such that $f = f' \circ g$. These over-categories assemble into a functor
\begin{aligned}
\mathcal{C} \to \operatorname{Cat}: c \mapsto \mathcal{C}/x
\end{aligned}
to the (large) category of small categories. Indeed, for a morphism $h: x \to x'$, we get a functor $\mathcal{C}/x \to \mathcal{C}/x': (f: y \to x) \mapsto (h \circ f: y \to x')$.
For every $x$, there is a canonical functor source functor $s: \mathcal{C}/x\to \mathcal{C}$ given by sending $f: y \to x$ to its source $y$. Essentially by definition, these source functors assemble into a cocone over $\mathcal{C}$ in $\operatorname{Cat}$, and thus give rise to a functor
\begin{aligned}
\operatorname{colim}_{c\in \mathcal{C}}\mathcal{C}/x \to \mathcal{C}.
\end{aligned}
This is the canonical functor referred to in the statement above.
Attempt: one can try to simply write down an inverse functor $\mathcal{C} \to \operatorname{colim}_{c\in \mathcal{C}}\mathcal{C}/x$ as follows. An object $x$ is sent to the object in the colimit system represented by the object $(\operatorname{id}_x: x \to x)$ in $\mathcal{C}/x$. If $f: x \to x'$ is a morphism in $\mathcal{C}$, notice that this object $(\operatorname{id}_x: x \to x)$ gets identified in the colimit system with the object $(f: x \to x')$ in $\mathcal{C} / x'$. Also notice that $f: x \to x'$ forms a morphism in $\mathcal{C}/x'$ from $(f: x \to x')$ to $\operatorname{id}_{x'}: x' \to x'$. It thus seems that the assignment $x \mapsto [\operatorname{id}_x: x \to x]$ extends to a functor $\mathcal{C} \to \operatorname{colim}_{c\in \mathcal{C}}\mathcal{C}/x$. I think this should be the desired inverse.
Question: Does the above proof attempt work? If so, is there a more invariant way of proving that the canonical functor is an equivalence? I am interested in a version of the statement for $\infty$-categories, for which one cannot simply write down an inverse functor by giving what it does on objects and morphisms.
Best Answer
Apparently I confused myself. The claim is true for both the strict colimit and the pseudo colimit, as you realised.
Consider the codomain projection $\operatorname{codom} : [\mathbf{2}, \mathcal{C}] \to \mathcal{C}$. This is a Grothendieck opfibration whose fibres are the slice categories $\mathcal{C}_{/ x}$ as $x$ varies, and the opcartesian morphisms are the ones whose domain part is an isomorphism. There is a canonical splitting, exhibiting this as the Grothendieck construction of the diagram $x \mapsto \mathcal{C}_{/ x}$ you are interested in. It is a general fact that the Grothendieck construction of a diagram is its oplax colimit, and one obtains the pseudo colimit by inverting the opcartesian morphisms.
Proposition. The domain projection $\operatorname{dom} : [\mathbf{2}, \mathcal{C}] \to \mathcal{C}$ is a localisation functor.
Proof. The domain projection is the right adjoint of the functor $\mathcal{C} \to [\mathbf{2}, \mathcal{C}]$ that sends each object $x$ to $\textrm{id}_x : x \to x$. The latter is fully faithful, so the domain projection is a coreflector. But reflectors are automatically localisation functors, so we are done. ■
Corollary. The pseudo colimit of $x \mapsto \mathcal{C}_{/ x}$ is equivalent to $\mathcal{C}$ via the canonical cocone. ■
The strict colimit can be constructed in a similar way: instead of inverting the opcartesian morphisms, we force them to be identity morphisms.
I am informed that similar general results hold in the setting of quasicategories, so the pseudo colimit version should translate directly.