Groupoid completion of a topological category vs its homotopy category?

Question

Given a category $\mathcal{C}$ enriched in spaces, we can take the nerve (a simplicial space) and then geometric realization to get a space $B\mathcal{C}$. If we view spaces as $\infty$-groupoids, then this process should be thought of as a ($\infty$-)groupoidification.

We can also consider the homotopy category $h\mathcal{C}$, which has the same objects as $\mathcal{C}$ but where the morphisms from x to y are given by $\pi_0 \mathcal{C}(x,y)$. This is an ordinary category and we can take the nerve and geometrically realize it to get the classifying space $Bh\mathcal{C}$.

In general the spaces $B\mathcal{C}$ and $Bh\mathcal{C}$ will be very different, but they might agree on some low dimensional homotopy groups.

Fix an object $x \in \mathcal{C}$. Is it true that $\pi_1(B\mathcal{C}, x)$ is isomorphic to $\pi_1( Bh\mathcal{C}, x)$? If not, what is a good counter example? Are there conditions under which these will be isomorphic? For example I am interested in the case where $\mathcal{C}$ is symmetric monoidal and $x$ is the unit object.

Note that we can view a set as a discrete topological space and so $h\mathcal{C}$ is also a (discrete) topological category. There is a functor $\mathcal{C} \to h\mathcal{C}$, and so there is a natural comparison map $\pi_1(B\mathcal{C}, x)\to\pi_1( Bh\mathcal{C}, x)$.

Answer 1

The $\pi_0$ and $\pi_1$ are the same. The former is obvious since, taking homotopy categories and groupoidifying do not affect connected components.

The fundamental group of an infinity category $S$ by van Kampen has a generator and relation description in terms of the 1 and 2 simplices. In particular, it has generators given by strings of 1-simplices and formal inverses that start and end at $*$ subject to the relation that we can exchange homotopic simplices, and that $ee^{-1}=e^{-1}e=Id$.

This group is the same as the group where we pick a single representative 1-simplex in each homotopy class and add in all the relations involving only these representatives.

Again by van Kampen, this group is exactly the fundamental group of the realization of $Ho(S)$, since we have just named each path component of the morphism space.

Then we simply transfer this back to topologically enriched categories and we are done.

This is the best one can hope for in general, since if $S$ is a Kan complex, its homotopy category is a groupoid and the realization of this is a 1-type (in particular the map you describe is the Postnikov approximation map).