Online Talk

By a relative category we mean a category $\mathcal{C}$ equipped with a class $\text{we}$ of weak equivalences.  Given such a thing, one can construct a simplicial set $N\mathcal{C}$, called the relative nerve.  (In the case where $\text{we}$ is just the class of identity morphisms, this is just the usual nerve of $\mathcal{C}$.)  Under mild conditions on $\mathcal{C}$, one can show that $N\mathcal{C}$ is a quasicategory, and that the homotopy category of $N\mathcal{C}$ is the category of fractions $\mathcal{C}[\text{we}^{-1}]$.  There is some existing literature on this, which we will review briefly; much of it depends on rather abstract methods involving different model structures on various categories.  Our main contribution will be to explain how some of this abstraction can be replaced by analysis of the combinatorial homotopy theory of certain finite posets.