se* as ‘*Most concepts and results of category theory can be extended to quasi-categories*’, [@joyal].

It would have been nice to have explored Joyal’s work on quasi-categories more fully, e.g. [@joyal], but time did not allow it. The following few sections just skate the surface of the theory.

Quasi-categories
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Categories yield quasi-categories via the nerve construction. Quasi-categories yield categories by a ‘fundamental category’ construction that is left adjoint to nerve. This can be constructed using the free category generated by the 1-skeleton of $A$, and then factoring out by a congruence generated by the basic relations : $gf \equiv h$, one for each commuting 1-sphere $(g,h,f)$ in $A$. By a *1-sphere* is meant a map $a : \partial \Delta[2] \to A$, thus giving three faces, $(a_0,a_1,a_2)$ linked in the obvious way. The 1-sphere is said to be *commuting* if there is a 2-simplex, $b\in A_2$, such that $a_i = d_ib$ for $i = 0,1,2$.

This ‘fundamental category’ functor also has a very neat description due to Boardman and Vogt. (The treatment here is adapted from [@joyal].)

We assume given a quasi-category $A$. Write $gf \sim h$ if $(g,h,f)$ is a commuting 1-sphere. Let $x,y \in A_0$ and let $A_1(x,y) 