The categories $\mathbf{Top}$ of topological spaces, $\mathbf{sSet}$ of simplicial sets and $\mathbf{Cat}$ of small categories are all equipped with a functor $\pi_0$ into the category $\mathbf{Set}$ of sets, which is a left adjoint and measures the number of connected components. There are also plenty of functors between the upper categories, which gives rise to the question, if they preserve the number of connected components.
I have already proven this for $|-|\colon\mathbf{sSet}\leftrightarrows\mathbf{Top}\colon \operatorname{Sing}$, for $\tau\colon\mathbf{sSet}\leftrightarrows\mathbf{Cat}\colon N$ and for $\operatorname{Sd}\colon\mathbf{sSet}\leftrightarrows\mathbf{sSet}\colon\operatorname{Ex}$. The functor $\operatorname{Ex}^\infty=\varinjlim_{n\in\mathbb{N}}\operatorname{Ex}^n\colon\mathbf{sSet}\rightarrow\mathbf{sSet}$ still works fine since the colimit commutes with the left adjoint $\pi_0$. The problem is, that $\operatorname{Sd}^\infty=\varprojlim_{n\in\mathbb{N}}\operatorname{Sd}^n\colon\mathbf{sSet}\rightarrow\mathbf{sSet}$ does not work fine because of the limit and since it is not the left adjoint of $\operatorname{Ex}^\infty$ and also writing either $X$ or $\operatorname{Sd}^n(X)$ as a colimit doesn't seem to work out. Does $\operatorname{Sd}^\infty$ also preserve the number of connected components? If yes, how is it proven and if no, is there a counterexample?
The fact, that there are many other categories with a forgetful functor to $\mathbf{Set}$ and left adjoints (free object functor) or right adjoints (cofree object functor) like we have for the three categories above, gives also rise to some more questions: Are there more categories with a functor like $\pi_0$ (like for example the category $\mathbf{Graph}$ of graphs) giving the number of connected components and functors to other categories with one, which I have not yet considered? If yes, is the number of connected components preserved? Are there other categories with functors similar to $\pi_0$ in the sense that they measure something similar, so we can look at more possible preservations?
Best Answer
Ignoring issues with what TOP really should be, let me focus on the question whether $\pi_0(Sd^\infty(X))=\pi_0(X)$ for a simplicial set $X$. If we look at $X=\Delta^1$, we should get a counterexample. Here $\pi_0$ should denote the equivalence class of vertices, where two vertices are equivalent, if there is a finite path (a list of one simplices ignoring the direction) joining them. In the inverse limit, we have uncountably many zero simplices, but only countably many (non-degenerate) one simplices. Thus $\pi_0(Sd^\infty(\Delta^1))$ must be uncountable.
A long time ago I thought a bit about these situations. Morally, I think there category of sets is somehow the wrong category for such questions. We can view any set as a discrete topological space. But then that functor $SET\to TOP$ is not compatible with limits.
I believe I could show a statement of the form that if $(X_n)_{n\in \mathbb{B}}$ is a inverse system of m-dimensional, simplicial (totally disconnected compact Hausdorff-spaces), then the canonical map $|\lim_n X_n|\to \lim_n |X_n|$ is a homeomorphism, where $m$-dimensional means that $Sk^m(X_n)\to X_n$ is a homeomorphism for all $n$ and $|-|$ is the non-fat geometric realization (the one that people would call the wrong one).
My motivation was just to write the Hawaiian earrings or compatifications of a tree as the geometric realization of something. That seemes to work, as long as we also allow a topology on the $n$-simplices. Funnily then one can also write $\Delta^1$ as the geometric realization with the Cantor-Set as its Zero-Skeleton.
I haven't published that but I might look it up if there is interest.