## What is a graded manifold?

There are several variations on the definition of “graded manifold”, and it can be a bit confusing for a newcomer. The goal of this post is to describe one definition that rules them all. This definition hopefully sheds some light on the relationships between the various types of graded manifolds.

I thought I had seen the definition in Kostant’s early treatise “Graded manifolds, graded Lie theory, and prequantization,” but, looking back at the paper now, I can’t find the definition there. In any case, I don’t claim any originality here. If anyone can remind me of the correct reference, I would appreciate it.

Let $S$ be a commutative monoid equipped with a homomorphism $\epsilon: S \to \mathbb{Z}_2$. We may say that an element $s \in S$ is even if $\epsilon(s) = 0$ or odd if $\epsilon(s) = 1$. Note that the homomorphism requirement simply means that the odd and even elements of $S$ behave the way we would expect them to: odd plus odd is even, odd plus even is odd, etc. (and also 0 is even).

An $S$-graded vector space is a vector space $\mathcal{V}$ (I’m going to assume we’re working over $\mathbb{R}$, but any field not of characteristic 2 will do) equipped with a decomposition $\mathcal{V} = \bigoplus_{s \in S} V_s$ into subspaces labeled by elements of $S$. If a vector $v$ is in $V_s$, then we may say that $v$ is homogeneous of degree $s$. Some authors also use the term weight.

An $S$-graded algebra is an algebra $\mathcal{A}$ which is $S$-graded as a vector space, and for which $\mathcal{A}_s \mathcal{A}_t \subseteq \mathcal{A}_{s+t}$. An $S$-graded algebra is graded-commutative if $ab = (-1)^{\epsilon(s) \epsilon(t)} ba$ for all $a \in \mathcal{A}_s$, $b \in \mathcal{A}_t$.

For each function $m: S \to \mathbb{N}$ with finite support, we define the $m$-dimensional graded coordinate space to be the sheaf $\mathcal{O}_m$ over $\mathbb{R}^{m(0)}$, defined by

$\mathcal{O}_m(U) = C^\infty(U) \otimes \left( \bigotimes_{\mathrm{odd}\; s} \bigwedge \mathbb{R}^{m(s)}\right) \otimes \left( \bigotimes_{\mathrm{even}\; s \neq 0} \mathbf{S} \mathbb{R}^{m(s)}\right)$

for any open set $U \subseteq \mathbb{R}^{m(0)}.$ This might look a bit intimidating, but it just means that you take $C^\infty(U)$ and you add $m(s)$ variables for each nonzero element of $s$, where the variables associated to odd elements of $s$ anticommute with each other, and the variables associated to even elements commute.

We view $\mathcal{O}_m$ as a sheaf of graded-commutative $S$-graded algebras, where the variables associated to $s$ are of degree $s$ (and the elements of $C^\infty(U)$ are of degree 0).

We’re finally ready to state the definition: An $S$-graded manifold of dimension $m$ is a topological space $M$ equipped with a sheaf $\mathcal{C}_\mathcal{M}$ of $S$-graded algebras which are locally isomorphic to $\mathcal{O}_m$. It may not be obvious, but the underlying space $M$ automatically becomes a manifold.

### Examples

For different choices of $S$ and the homomorphism $\epsilon$, we can obtain various types of graded manifolds.

• If $S = \mathbb{Z}_2$ and $\epsilon$ is the identity map, then you get supermanifolds.
• If $S = \mathbb{Z}$ and $\epsilon$ is the “mod 2” map, then you get the $\mathbb{Z}$-graded manifolds that I studied in my thesis.
• If $S = \mathbb{Z} \times \mathbb{Z}_2$ and $\epsilon$ projects onto the $\mathbb{Z}_2$-component, then you get Ted Voronov’s graded manifolds. The key point here is that the $\mathbb{Z}$-component of the grading is completely unrelated to the property of begin even or odd.
• If $S = \mathbb{N}$ and $\epsilon$ is the “mod 2” map, then you get the N-manifolds that were emphasized by Severa and Roytenberg.

Of course, these are only a few of the many possibilities; for example, this paper by Grabowski and Rotkiewicz considers $\mathbb{N}^n$-gradings.

### Relations

Let $S$ and $S'$ be commutative monoids as above. Then any homomorphism $\phi: S \to S'$, respecting the maps to $\mathbb{Z}_2$, induces a functor from the category of $S$-graded manifolds to that of $S'$-graded manifolds.

If you’re willing to ignore technical details, the idea is that $\phi$ induces a functor at the level of graded algebras, which may then be applied to the sheaves. The technical issue is that the underlying topological spaces change when there are coordinates whose degree is in the kernel of $\phi$. But everything can be made to work nonetheless.

For example, we have that for any $S$, there is a functor from the category of $S$-graded manifolds to that of supermanifolds, induced by $\epsilon$ itself.

But, for my purposes, the main example is the functor taking $\mathbb{Z}$-graded manifolds to Voronov’s graded manifolds, induced by the map $\mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}_2$, $k \mapsto (k,[k])$. It can be shown that, if $\phi$ is injective, then the induced functor is injective and fully faithful. So, in this example, we see that $\mathbb{Z}$-graded manifolds form a full subcategory of Voronov’s graded manifolds.

I hope this brief introduction was helpful. Please comment if there are any details or further issues you’d like me to clarify in a future post.