If $\mathcal{L}$ is the $\mathbf{sSet}$-enriched subcategory of $\mathbf{sSet}$ whose objects are finite coproducts of the terminal simplicial set $\Delta^0 = \Delta(-,[0]) = *$, identify the object $\mathcal{L}(*,*)$.

One obvious candidate for $\mathcal{L}(*,*)$ is $\Delta^1$. To that end, we know that $1_{*} = d_o \circ s_0 = d_1 \circ s_0$ by the simplicial identities.

Another (perhaps more likely) canditate for $\mathcal{L}(*,*)$ is $*$ itself, i.e., $\mathcal{L}(*,*) \cong *$. To that end, we know that there exists a unique morphism $1_{\mathcal{L}(*,*)}: \mathcal{L}(*,*) \rightarrow *$, and that there exists as morphism $id: * \rightarrow \mathcal{L}(*,*)$ via the enrichment structure.

  • $\begingroup$ If $\bf sSet$ is regarded as enriched over itself, and the question is "what is ${\cal L}(*,*)$?", then it should be easy to prove that ${\cal L}(*,*)$ is the terminal simplicial set. $\endgroup$ – Fosco Loregian Oct 26 '18 at 21:58
  • $\begingroup$ It should be, but the direct proof is escaping me. It seems like this would be some sort of general proof that, for a general $\mathcal{V}$-category $\mathcal{C}$, we would have $\mathcal{C}(*,*) \cong *$, but the proof is eluding me. $\endgroup$ – Joseph R Denman Oct 29 '18 at 1:40

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