show/hide this revision's text 2 Added two examples.

Examples

I just want to amend the above discussion with two more examples where this notion of "cosupport" naturally appears. First, consider the a projective family of finite dimensional manifolds $X_k \to X_l$ with $k\ge l$, indexed by $\mathbb{N}$ (for example, $X_k = J^k(M,N)$ is the jet manifold of maps from $M$ to $N$). The categorical limit (taken in a sufficiently general category) an pro-finite dimensional manifold $X$ equipped with natural projections $X\to X_k$ (for instance $X=J^\infty(M,N)$). The smooth function $C^\infty(X)$ consist of those that correspond to pullbacks of elements of the $C(X_k)$'s along the natural projections. In coordinate language, a smooth function on $X$ depends smoothly on finitely many of its infinitely many coordinates. I think it would be natural to compress this statement as follows: $C^\infty(X)$ consists of smooth functions of finite cosupport.

On the other hand, consider finite dimensional manifolds $M$ and $N$. The mapping space $X=C^\infty(M,N)$ can be given the structure of a Fréchet manifold. If we consider the inclusions $M_i \subseteq M$ of all compact subsets with open interior, as well as the inclusions between $M_i \subseteq M_j$ when they exist, we can consider $M$ as the categorical colimit of the resulting inductive system in the manifold category. The mapping spaces, on the other hand, form a projective system $C^\infty(M_j,N) \to C^\infty(M_i,N)$ with pullbacks of the inclusions as morphisms. The categorical limit of this projective system is just the global mapping space $X = C^\infty(M,N) \to C^\infty(M_i,N) = X_i$ with corresponding natural projections. Now, there is an important subalgebra of the algebra of smooth functions on $X$, $C^\infty_{cc}(X) \subseteq C^\infty(X)$, consisting of the images of the pullback maps $C^\infty(X_i) \to C^\infty(X)$ induced by the natural projections $X \to X_k$. I think it would be natural to call $C^\infty_{cc}(X)$ the algebra of smooth functions of "compact cosupport". (Incidentally, this algebra has appeared recently in the literature on the rigorous construction of classical field theories in the mathematical physics literature.)
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Terminology for notion dual to "support"

If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\in X$ such that $f(i) \ne 0$. I think this usage of the word support is more or less standard. On the other hand, consider the vector space $V = \mathbb{R}^X$ consisting of all real functions on $X$. It has a preferred basis $\delta_i$, consisting of characteristic functions on singletons $\{i\}$. So, any function on $X$ can be written $f = \sum_i x^i \delta_i$, such that $f(i) = x^i$. The $x^i$ are then preferred coordinates on $V$. Finally, consider a function $F$ on $V$ that depends only on some of the coordinates. That is $F=F(\{x^i\}_{i\in S})$, where the coordinates on which it depends are indexed by a subset $S\subseteq X$.

Question: Is there a name for this subset $S\subseteq X$ in relation with $F$ as a function on $V=\mathbb{R}^X$? It is very tempting to call it some kind of support of $F$, but that terminology conflicts with the support of $F$ as a function on $V$.

Now, on to some categorical hand waving to say why I think the role of $S$ in my question is somehow dual to the notion of support I defined in the first paragraph. The set $X$ is an object in the category $\mathbf{Set}$. This object happens to be the colimit of the subcategory indexed by $\mathcal{P}(X)$, the subsets of $X$, with natural injections between those subsets that contain each other. On the other hand, the vector space $V=\mathbb{R}^X$ is an object in the category $\mathbf{Vect}$, but can also be considered an object in $\mathbf{Set}$ after applying the appropriate forgetful functor. The contravariant functor $\mathbb{R}^(-)$ maps $X$ to $V$ and the diagram indexed by $\mathcal{P}(X)$ to another diagram with all arrows reversed, whose limit (in either $\mathbf{Vect}$ or $\mathbf{Set}$) gives $V$.

So, we have two diagrams in $\mathbf{Set}$, both indexed by $\mathcal{P}(X)$ (one covariantly and one contravariantly). The colimit of one diagram is $X$ and the limit of the other diagram is $V$. This setup gives a categorical characterization of the support of a function $f$ on $X$. If $S=\operatorname{supp} f$, then $f$ is in the image of the push-forward map $\mathbb{R}^S\to \mathbb{R}^X$ induced by the inclusion injection $S\to X$, where push-forward is extension by $0$. The subset $S$ is the support of $f$ because it is the smallest such subset. Thus, the construction of $X$ as a colimit is tightly connected with the notion of support.

On the other hand, if $F$ is a function on $V$, as in the question above, and $S$ the set defined therein, then $F$ is in the image of the pull-back map $\mathbb{R}^{\mathbb{R}^S} \to \mathbb{R}^V$ induced by the projection $V\to \mathbb{R}^S$ in the diagram whose limit is $V$. Again, $S$ is the smallest subset of $X$ with this property. So, it seems to me, that the relation of $S$ to $F$ and $V$ is in a sense dual to the relation of $\operatorname{supp} f$ to $f$ and $X$, since the former uses the limit structure of $V$ whereas the latter uses the colimit structure of $X$. Unfortunately, the fact that the two diagrams are both indexed by $\mathcal{P}(X)$ makes things a bit confusing. But, could $S$ be justly called the cosupport of $F$?

The same story could be told in the category of topological space $\mathbf{Top}$ instead of $\mathbf{Set}$. A topological space $X$ can then be seen as a colimit of its closed subsets. Applying the above categorical definition, we recover the notion of the support of a continuous function $f$ on $X$. If $V=C(X)$ and $F$ is a continuous linear functional on $V$, that is, a distribution, the set $S$ is also called the support of $F$. However, once we start considering non-linear functions on $V$, the set $S$ does not seem to have a convenient name. Finding such a convenient name was actually the original motivation for this question.