# $\mathbf{sSet}$-enriched Algebraic Theories

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.

• 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. – Fosco Loregian Oct 26 '18 at 21:58
• 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. – Joseph R Denman Oct 29 '18 at 1:40