# What is the fundamental category?

Given a category $$\mathcal{C}$$, we have a nerve functor $$\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$$ that assigns to $$\mathcal{C}$$ its nerve $$\mathrm{N}(\mathcal{C})$$. This functor seems to have a left adjoint $$\tau_1 \colon \mathbf{Set}_{\Delta} \to \mathbf{Cat}$$ that assigns to a simplicial set $$X$$ its fundamental category, as in Joyal's Notes on Quasi-Categories.

There it also states that the fundamental grouped $$\pi_1 X$$ is obtained by inverting the arrows of $$\tau_1 X$$, but there is no construction of $$\tau_1 X$$.

What is the construction/definition of the fundamental category of a simplicial set $$\tau_1 X$$? What are its objects and morphisms?

The best presentation that I know of is in Riehl and Verity: 1.1.10&11

• Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently – user313212 Dec 28 '18 at 17:38
• Yes, as proved in 1.1.11. – Ivan Di Liberti Dec 28 '18 at 17:40