Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

6.1 Expanded Tags

The tags in this section were retired because they contained an explanation which was later expanded upon.

Example 6.1.0.1 (Simplicial Sets of Dimension $\leq 0$). The contents of this tag have been expanded to subsection §1.1.4.

Let $S_{\bullet }$ be a simplicial set. Then the $0$-skeleton $\operatorname{sk}_0( S_{\bullet } )$ can be identified with the coproduct $\coprod _{v \in S_0} \{ v\} $, indexed by the collection of all vertices of $S_{\bullet }$. In particular, the simplicial set $S_{\bullet }$ has dimension $\leq 0$ if and only if it is isomorphic to a coproduct of copies of $\Delta ^0$. We therefore obtain an equivalence of categories

\[ \xymatrix { \{ \text{Simplicial Sets of Dimension $\leq 0$} \} \simeq \{ \text{Sets} \} . } \]

Example 6.1.0.2. The contents of this example have been expanded to §1.2.6.

Let $G$ be a directed graph (Definition 1.1.5.1) and let $S_{\bullet }$ denote the associated simplicial set of dimension $\leq 1$ (Proposition 1.1.5.9). Then the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ can be described explicitly as follows:

  • The objects of $\mathrm{h} \mathit{S}_{\bullet }$ are the vertices of the graph $G$.

  • Given a pair of vertices $v,w \in \operatorname{Vert}(G)$, a morphism from $v$ to $w$ in $\mathrm{h} \mathit{S}_{\bullet }$ is given by a path from $v$ to $w$ in the directed graph $G$: that is, an ordered sequence of edges $(e_1, e_2, \ldots , e_ n)$ satisfying $s( e_1 ) = v$, $t(e_ n) = w$, and $t( e_ i ) = s( e_{i+1} )$ for $0 < i < n$. Here $s,t: \operatorname{Vert}(G) \rightarrow \operatorname{Edge}(G)$ denote the source and target maps. Moreover, we allow $n=0$ in the case $v = w$ (the empty sequence is regarded as the identity morphism from the vertex $v$ to itself).

  • Composition of morphisms in $\mathrm{h} \mathit{S}_{\bullet }$ is given by concatenation of paths. More precisely, given morphisms $f = (e_1, e_2, \ldots , e_ m)$ from $u$ to $v$ and $g = (e'_1, e'_2, \ldots , e'_ n)$ from $v$ to $w$, the composition $g \circ f$ is given by the sequence $(e_1, e_2, \ldots , e_ m, e'_1, \ldots , e'_ n)$.