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Domenicos comment lead to the following idea (I am posting this as an answer, as it is too long for a comment): Let $CAT$ denote the category of small categories and $CAT'$ denote the category, whose …

The geometric realization of a simplicial set and the geoemetric realization of its barycentric subdivision are always homeomorphic. However there cannot be a natural isomorphism between these two fu …

I believe surjective on singletons is neccessary as well. For this we have to show that a relation $f:X\to Y$ that is not surjective on singletons is not an epimorphism. Let $a\in Y$ be a point such t …

Let us compare what both functors do on $X=S^2=\Delta^2/\partial \Delta^2$. There are only two nondegenerate simplices, so the first functor sends it to an interval. The second functor sends it to the …

Let $R$ be a ring. Consider the inclusion functor from the category of finitely generated, projective $R$-modules to the category of all finitely generated $R$-modules. For which rings does it have a …

Ok I checked the idea mentioned in the comments above. As $X_n$ is a closed subspace of $X_M$ (use any degeneracy; it has a left inverse; composing both the other way round yields a projection and pr …

I think, that the levelwise colimit agrees with the colimit in the category Func$(\Delta,\mathcal{C}$. As the colimit may be viewed as a functor from the category of directed systems over $\mathcal{C} …

I see one example with 7 morphisms. It is a subcategory of the category of groups. The only object is the the Klein 4-group $(\mathbb{Z}/2)^2$, and the morphisms are generated by the two projections a …

Let $f:X \rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from …

Suppose $X$ is a simplicial space of dimension $M$ (i.e. all simplices above dimension $M$ are degenerate). The claim is: $|X|$ is compact. iff $X_n$ is compact for each $n$. Suppose each $X_n$ is c …

Wikipedia already gives a list of some examples.

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 …

Let us use increasing filtrations $F^pC \subset F^{p+1}C\subset \ldots$. Then the homology of $C$ and $C'$ also inherit the structure of filtered modules by $F^pH(C) = im(H(F^p(C)\rightarrow C))$, or …

Suppose we have two objects $X,Y$ with a $G$-operation and a non-equivariant map between them. In this situation, we can look at the largest subobject $X'$ of $X$ on which $f$ is $G$-equivariant. Is t …