A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is the category of all groups and homomorphisms. For examples, consider the homology groups $H\_{n}(X)$ or the homotopy groups $\pi\_{n}(X)$. Of course, a problem with these invariants is that they are not fully faithful functors, i.e., $H\_{n}(X)\cong H\_{n}(Y)$ does not imply that $X$ and $Y$ are homeomorphic. The existence of a fully faithful functor $F:\mathscr{T}\to\mathscr{G}$ would imply that $\mathscr{G}$ has a subcategory $F\mathscr{T}$ equivalent to $\mathscr{T}$. This would be both rather disturbing and extremely interesting. First, it would mean that in a sense, all of topology is just a subset of group theory, which would be rather disturbing to topologists, but it would also reveal a fundamental connection between two seemingly disparate disciplines. My question is: is it possible to prove that no such functor exists? In other words, could one exhibit some categorical property that $\mathscr{T}$ posesses that $\mathscr{G}$ does not. This question can naturally be extended to other important categories, like $\mathscr{M}$, the category of all modules, or $\mathscr{R}$, the category of all rings. So in general, given arbitrary categories $\mathscr{C}$, $\mathscr{D}$, is there any natural way of showing that no fully faithful functor $F:\mathscr{C}\to\mathscr{D}$ exists, i.e., are there any nice "categorical invariants?"


EDIT: A couple people pointed out that I really ought to be discussing fully faithful functors, rather than just faithful functors. Also, I have changed the title in accordance with Martin Brandenburg's recommendation.



Proving the impossibility of an embedding of categories
One property that $Top\_{cgwh}$ has (if we let $Top\_{cgwh}$ be the full subcategory of $Top$ consisting of compactly generated, weakly Hausdorff spaces) that $Grp$ doesn't is cartesian closedness: the hom-set $Grp(G,H)$ is not a group. Another property that $Top$ has that $Grp$ doesn't is an initial object distinct from its final object, but these are just off the top of my head. 


But thinking of these functors as being intrinsic to $Top$ is a small mistake, because $Top$ is just a presentation of the $(\infty,1)$-category of homotopy types, and these functors are really giving invariants of homotopy types.


This doesn't really answer your question, but I hope it indicates that the sort of functor you are looking for won't be anything like cohomology or homotopy.




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Edit: what about the composite $Top \to Set \to Grp$ where the first is the 'underlying set' functor and the second is the 'free group' functor? This is faithful, but perhaps not very useful for topological purposes. (Edit again: I just noticed that Tom Goodwillie also pointed out a similar example)


