How does the forgetful functor from $\textsf{Man}$ to $\textsf{TopMan}$ factor under the ternary factorization system?

Question

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

Answer 1

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.