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Here, $\textsf{TopMan}$ refers to the category of topological manifolds (second-countable, Hausdorff, locally Euclidean topological spaces) with continuous maps for morphisms; and $\textsf{Man}$ refers to the category of smooth manifolds (topological manifolds equipped with a smooth atlas) with smooth maps for morphisms.

Clearly, $\textsf{TopMan}$ forms a full subcategory of $\textsf{Top}$. More importantly, the following are true (please correct me if I'm wrong at any step):

  1. Every smooth manifold has a unique underlying topological manifold and every smooth map of manifolds is continuous. Thus, the "forgetful" functor $F: \textsf{Man} \rightarrow \textsf{TopMan}$ is well-defined.

  2. Not every topological manifold admits a smooth structure (due to Kervaire). Thus, $F$ is not essentially surjective.

  3. $F$ is not full because of the much simpler reason that not all continuous maps (between smooth manifolds) are smooth.

  4. A given topological manifold may admit more than one smooth structure (due to Milnor). Thus, $F$ is not injective-on-objects.

  5. $F$ is indeed faithful, because given any two smooth manifolds $X$ and $Y$, then two distinct smooth maps between $f,g:M \to N$ cannot have the same image under $F$ (in fact, $f$ and $g$ are distinct iff they are distinct as set-functions.)

Given all this, how do I factorize $F$ in the 3-way factorization system [1] [2]? Since, $F$ is faithful, it only needs to be written as a composition of two functors: one that is fully faithful, and another that is essentially surjective and faithful.

While all this isn't particularly essential/important to the study of $\textsf{Man}$, I think that thinking of the forgetful functor in terms of its factors will help me get a feel for what exactly smooth manifolds and smooth maps are in terms of structure/property.

PS: I have been unsuccessful so far in locating online discussions on $\textsf{Man}$ and related categories (beyond the most basic stuff). Any links will be appreciated (or you can tell me why talking about this category as such is not very useful)!

[1] http://ncatlab.org/nlab/show/stuff,+structure,+property

[2] http://ncatlab.org/nlab/show/ternary+factorization+system

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The ternary factorisation of a functor $F : \mathcal{C} \to \mathcal{D}$ can be constructed explicitly as follows:

  • Let $F[\mathcal{C}]$ be the category whose objects are those in $\mathcal{C}$ and whose morphisms $X \to Y$ are the morphisms $F X \to F Y$ in $\mathcal{D}$ that are in the image of $F : \mathcal{C} (X, Y) \to \mathcal{D} (F X, F Y)$.
  • Let $\mathcal{D}'$ be the category whose objects are those in $\mathcal{C}$ and whose morphisms $X \to Y$ are the morphisms $F X \to F Y$ in $\mathcal{D}$.
  • Then the functor $F : \mathcal{C} \to \mathcal{D}$ has an evident factorisation $\mathcal{C} \to F [\mathcal{C}] \to \mathcal{D}' \to \mathcal{D}$ where $\mathcal{C} \to F [\mathcal{C}]$ is bijective on objects and full, $F [\mathcal{C}] \to \mathcal{D}'$ is bijective on objects and faithful, and $\mathcal{D}' \to \mathcal{D}$ is fully faithful.

For your example where $F : \mathcal{C} \to \mathcal{D}$ is the forgetful functor $\mathbf{Man} \to \mathbf{TopMan}$, $F [\mathcal{C}] = \mathbf{Man}$ and $\mathcal{D}'$ is the category of smooth manifolds and continuous maps.

Note also that there are other ternary factorisation systems on $\mathbf{Cat}$. For instance, we can factor every functor as a bijective on objects full functor followed by a surjective on objects faithful functor followed by an injective on objects fully faithful functor.

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  • $\begingroup$ Thank you; that was very clearly explained. But, just to make sure, in your scheme, going from D' to D would forget both the "structure" (i.e. it strips the atlas) and the "property" (i.e it adds non-smoothable manifolds) in the same step - isn't it? (I've always been told category theory is about morphisms, not objects - so perhaps I've interpreted "structure" and "property" wrongly) $\endgroup$ – udit.m Oct 22 '15 at 10:30
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    $\begingroup$ The structure is already forgotten when we get to $\mathcal{D}'$. (Don't get confused by the "labelling" of the objects – as you say, it is the morphisms that are important.) $\endgroup$ – Zhen Lin Oct 22 '15 at 10:35
  • $\begingroup$ Wait, I'm confused. In $\mathcal{D'}$, are two non-diffeomorphic smooth manifolds (with the same underlying topological space) identified as one object or not? If they are identified as one object, then $F[ \mathcal{C} ] \to \mathcal{D'}$ would be surjective on objects, not bijective on objects (refer answer); and as you say, structure is already forgotten. If they aren't identified, of course, it is bijective on objects, but structure is not forgotten (for, how else would the two smooth structures be distinct?). $\endgroup$ – udit.m Oct 22 '15 at 11:03
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    $\begingroup$ If you have two smooth structures on the same topological manifold, then you have isomorphic objects in $\mathcal{D}'$. Therefore they cannot be said to have different structures as objects in $\mathcal{D}'$. $\endgroup$ – Zhen Lin Oct 22 '15 at 11:14
  • $\begingroup$ Ah! Because homeomorphisms are the isomorphisms in $\mathcal{D'}$. Got it. Thanks! $\endgroup$ – udit.m Oct 22 '15 at 11:20

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