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Given a category $\mathcal{C}$ and an object $x\in \mathcal{C}$ we can look at the over category $\mathcal{C}_{/x}$ whose objects are morphisms $d \rightarrow x$ for $d\in \mathcal{C}$ and morphisms are commutative triangles.

In Higher Topos Theory, Lurie defines the overcategory $\mathcal{C}_{/x}$ in the following way: Let $[0]$ be the category with a single object and only the identity morphism. specifying an object of $\mathcal{C}$ is the same as specifying a functor $x:[0] \rightarrow \mathcal{C}.$ If $\mathcal{A,B}$ are categories, let $\mathcal{A \star B}$ denote their categorical join. Then we define $\mathcal{C}_{/x}$ via universal property:

$Hom(\mathcal{C^\prime},\mathcal{C}_{/x})\cong Hom_x(\mathcal{C^\prime}\star[0],\mathcal{C})$

where the subsript $x$ on the RHS means we only look at functors $\mathcal{C^\prime}\star [0] \rightarrow \mathcal{C}$ whose restriction to $[0]$ coincides with $x$.

We can generalize this to consider overcategories of a morphism.

More generally, Let $S,K$ be simplicial sets (which we can think of as categories) and $p:K \rightarrow S$ a simplicial map (which we can think of as a functor). Then there exists a simplicial set $S_{/p}$ with the following universal property:

$Hom_{Set_\Delta}(Y,S_{/p})=Hom_p(Y\star K, S)$

where the subscript on the RHS means we only consider morphisms $f:Y \star K \rightarrow S$ s.t. $f|K=p$.

I'm having a hard time understanding this definition. I want to look at a concrete example- let $K=[1].$ Let $f:x \rightarrow y$ be a 1-morphism in $S$. We have a simplicial map $f:[1] \rightarrow S$ where $f(0)=x,f(1)=y,$ and $f(0 \rightarrow 1)$ is the map $f:x \rightarrow y$ in $S.$ Then what is $S_{/f}$?

I was naively thinking the objects of this overcategory are commutative squares, i.e.

$\begin{array}{ccc} a & \overset{g}{\rightarrow} & b\\ \downarrow & & \downarrow\\ x & \overset{f}{\rightarrow} & y \end{array}$

Morphisms would be commutative triangular prisms over $f$. But is this the right way to think about overcategories? I'm afraid this intuition would fail if $K$ is a more complicated simplical set.

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Perhaps it would be better to call $\mathcal{C}_{/ f}$ the category of cones over $f$. The objects are commutative diagrams as below, $$\require{AMScd} \begin{CD} a @= a \\ @VVV @VVV \\ x @>>{f}> y \end{CD}$$ i.e. a cone over $f$, the morphisms are morphisms of cones, and so on.

Note that if $\mathcal{C}$ is an ordinary category then the canonical projection $\mathcal{C}_{/ f} \to \mathcal{C}_{/ x}$ is an isomorphism. Thus, it is perhaps not the best example to think about.

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