Global Polyfold Fredholm setup

EP-groupoid basics

A polyfold should be an "M-polyfold with isotropy". This is implemented via the language of EP-groupoids, which in finite dimensions reduce to orbifolds. First, recall that a groupoid is a small category in which every morphism is invertible (hence a groupoid with a single object is the same thing as a group). Now define an EP-groupoid to be a groupoid $X$ with morphism set ${\mathbf X}$ satisfying these properties:

• (Lie): $X$ and ${\mathbf X}$ are equipped with M-polyfold structures, with respect to which the source, target, multiplication, unit, and inversion maps

$s:{\mathbf X}\to X,\quad t:{\mathbf X}\to X,\quad m:{\mathbf X}{}_{s}\times _{t}{\mathbf X}\to X,\quad u:X\to {\mathbf X},\quad i:{\mathbf X}\to {\mathbf X}$

are sc-smooth.

• (etale): $s$ and $t$ are surjective local sc-diffeomorphisms.
• (proper): For every $x\in X$, there exists a neighborhood $V(x)$ so that $t:s^{{-1}}{\bigl (}\overline {V(x)}{\bigr )}\to X$ is proper.

Note that (Lie) makes sense because (etale) hypothesis implies that ${\mathbf X}{}_{s}\times _{t}{\mathbf X}$ inherits an M-polyfold structure. Moreover, (proper) implies that each isotropy group Failed to parse (lexing error): \mathbf G(x) := \{g \:|\: s(g) = t(g) = x\}

is finite.

We denote the orbit space by $|X|$. A polyfold structure on a (paracompact, Hausdorff) space $Z$ is simply $(X,\beta )$ where $X$ is an EP-groupoid and $\beta :|X|\to Z$ is a homeomorphism.

Example: We work out the EP-groupoid structure of $DM(1,2)$ in detail.

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