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Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small.

Since I suspect this category is not essentially small, I would be satisfied with knowing the answer to the question with "Fréchet manifold" replaced by "Fréchet manifold admitting smooth partitions of unity" (a relevant related question is Which Fréchet manifolds have a smooth partition of unity?). I would be likewise be happy to know it for metrisable Fréchet manifolds (which I believe to include those with smooth partitions of unity).

EDIT : I will add that my motivation is to see whether there is a sensible essentially small category of Fréchet manifolds including smooth mapping spaces, closed under natural operations (finite limits, when they exist, and reasonably-sized coproducts), which is defined by topological criteria not explicitly depending on cardinals. An instance of the latter would be something like asking that the underlying set is smaller than some given cardinal

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  • $\begingroup$ [Note to self...] The paper Henderson, David W. Infinite-dimensional manifolds are open subsets of Hilbert space. Bull. Amer. Math. Soc. 75 1969 759–762. claims that separable metrisable Fréchet topological manifolds are embeddable in the unique separable infinite-dimensional Hilbert space $H$ (e.g. $L^2([0,1])$), which class includes, if I'm not mistaken, smooth loop spaces. $\endgroup$
    – David Roberts ♦
    Commented Feb 13, 2016 at 2:18
  • $\begingroup$ I'm a bit suspicious though, since it claims that infinite-dimensional separable Fréchet spaces are homeomorphic to $H$, quoting Anderson, R. D. Hilbert space is homeomorphic to the countable infinite product of lines. Bull. Amer. Math. Soc. 72 1966 515–519. and I'm pretty sure that the space of smooth loops in $\mathbb{R}^n$ with the Fréchet topology is not a Hilbert space. $\endgroup$
    – David Roberts ♦
    Commented Feb 13, 2016 at 2:19

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Andrew Stacey points out elsewhere that arbitrary Hilbert spaces have smooth partitions of unity, so that criterion is not enough to get an essentially small category of Fréchet manifolds. For the purposes of considering mapping spaces between finite-dimensional manifolds, however, arbitrary Hilbert spaces are a bit of an overkill!

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