# Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces

Warmup:

Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric realization of it's nerve. The description $$\pi_0(B\mathcal{C})\cong\{\text{objects in }\mathcal{C}\}/\{\text{morphisms in }\mathcal{C}\}$$ is well known, but as I could not find a proof in the literature and as it will illustrates my question, a proof will follow:

Let $(x_0\rightarrow ...\rightarrow x_i,\lambda)\in N_{i}(\mathcal{C})\times\Delta^i$ be a representative of a point in $B\mathcal{C}$. As $\Delta^i$ is path-connected, we can connect $\lambda$ to $(1,0,...,0)$ and get a path from $[(x,\lambda)]$ to $[(x_0\rightarrow ...\rightarrow x_i,(1,0,...,0)]=[(x_0,1)]$. This shows that every path component has a 0-simplex as a representative. Now take a path $\gamma$ in $B\mathcal{C}$ which connects two points $[(x_0,1)], [(x_1,1)]$. Since $B\mathcal{C}$ is a CW-complex, the relative cellular approximation theorem yields a path between the same points in the 1-skeleton, but as $N_{1}(\mathcal{C})$ is discrete, it is constant. Thinking about which points of $N_1(\mathcal{C})\times\Delta^1$ got identified in the 1-skeleton, one concludes that there is a zig-zag of morphisms between $x_0$ and $x_1$.

The question:

Let $\mathcal{D}$ be a category internal to topological spaces, then it's nerve is a simplicial space and one can take the geometric realization of this simplicial space to obtain the topological classifying space $B\mathcal{D}$ of $\mathcal{D}$. Be aware of the fact, that this space does not have to be a CW-complex.

Now, is there a similar simple description of $\pi_0(B\mathcal{D})$ like in the discrete case? Maybe in terms of $\pi_0(Ob(\mathcal{D}))$ and $\pi_0(Mor(\mathcal{D}))$?

• How exactly is the quotient (objects)/(morphisms) being defined? Do you mean to indicate that two objects $x$ and $y$ are equivalent if there exists $f:x \to y$? Or maybe $f:y \to x$? Or either? May 15, 2015 at 14:40
• Neither: two objects become identified if there is a zigzag of morphisms connecting them. May 15, 2015 at 14:44
• Exactly. Compare ncatlab.org/nlab/show/connected+category
– Tom
May 15, 2015 at 14:45
• Morally, $\pi_0 (B \mathcal{D})$ should be the coequaliser of $\pi_0 (\operatorname{mor} \mathcal{D}) \rightrightarrows \pi_0 (\operatorname{ob} \mathcal{D})$; however, $\pi_0 : \mathbf{Top} \to \mathbf{Set}$ is not a left adjoint unless $\mathbf{Top}$ means something like locally connected spaces. May 15, 2015 at 14:53
• I would accept to restrict to categories $\mathcal{D}$ which objects and morphisms are locally connected or other convenient point-set topological restrictions, as long as all the constructions (taking pullbacks for the nerve, etc.) find place in the category of all topological spaces, since I don't want to get another space $B\mathcal{D}$ by changing the topological category.
– Tom
May 15, 2015 at 15:34

If everything takes place in the category of compactly generated spaces, it holds $$\pi_0(BC)=\pi_0(obC)/\tilde{},$$ where two path components in the object space get identified, if there are objects in each of them respectively which are connected by a morphism in the category and $\tilde{}$ is the equivalence relation generated by that.