Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

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 Fr├ęchet 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.)

share|improve this question
2  
I don't know if this point is important, but usually, the support of $f$ is the closure of the set on which $f$ is non-zero (assuming some topology on the domain of $f$). –  Dirk Oct 15 '12 at 20:13
1  
I do acknowledge this in my very last paragraph with the possibly obscure-to-many remark that a "topological space $X$ can then be seen as a colimit of its closed subsets." –  Igor Khavkine Oct 15 '12 at 21:21
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.