# Groupoid completion of a topological category vs its homotopy category?

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 $$\pi_1(B\mathcal{C}, x)\to\pi_1( Bh\mathcal{C}, x)$$.

• Alternative argument: The induced map $N_p(\mathcal{C}) \to N_p(h\mathcal{C})$ is $(2-p)$-connected for all $p$, which implies the map of (thick) geometric realizations is 2-connected (e.g. [arXiv:1403.2334, Prop 2.7]). – user168706 Nov 16 '20 at 18:32

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.

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).
• This does generalize to the higher homotopy groups in the sense that the homotopy category is a model for the infinity category where you replace the mapping space with its n-type for $n=0$. I believe in this context, we have isomorphisms on homotopy groups up to $n+1$, since these should be possible to achieve just by attaching $(n+3)$-simplices. – Connor Malin Jun 26 '20 at 2:15