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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 $\pi_0(Mor(\mathcal{D}))$?

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  • $\begingroup$ 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? $\endgroup$ – Vidit Nanda May 15 '15 at 14:40
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    $\begingroup$ Neither: two objects become identified if there is a zigzag of morphisms connecting them. $\endgroup$ – Zhen Lin May 15 '15 at 14:44
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    $\begingroup$ Exactly. Compare ncatlab.org/nlab/show/connected+category $\endgroup$ – Tom May 15 '15 at 14:45
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    $\begingroup$ 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. $\endgroup$ – Zhen Lin May 15 '15 at 14:53
  • $\begingroup$ 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. $\endgroup$ – Tom May 15 '15 at 15:34
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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.

This follows from Lemma 11.11 in Peter May's 'The Geometry of Iterated Loop Spaces', in which the more general claim for simplicial spaces is proven.

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