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?

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