# Defining Category of Problems

Let $\left\{\Pi_i\right\}_{i \in I}$ be a family of problems. Let problem $\Pi_i$ have solution $u_i$ lying in some solution space $X_i$. I am interested in making this set into a category. Is it meaningful to define morphisms between problems as follows?

Define a morphism $f_{ij}:\Pi_i \rightarrow \Pi_j$ if there exists a map $\sigma: X_i \rightarrow X_j$ such that $u_j = \sigma(u_i)$.

Any insights would be greatly appreciated.

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You are interested in making this set into a category... for what purpose? Certainly this is a sensible definition, but whether it will actually help you do anything depends on what you want to do. –  Qiaochu Yuan Nov 16 '11 at 15:22
I want to be able to define products of problems: decomposing a bigger problem to a set of smaller ones. –  Manos Nov 16 '11 at 15:46
This sounds a bit like the category of contexts. –  Zhen Lin Nov 16 '11 at 17:56
@Zhen: this is very interesting. –  Manos Nov 16 '11 at 18:05
@Henning: I am thinking of a well-defined mathematical class of problems. For example: the matrix inversion. In that case each $\Pi_i$ would be indexed by a nonsingular matrix. –  Manos Nov 16 '11 at 21:40

Your definition as it stands seems to be too liberal to be really useful; since it only depends on one particular value of $\sigma$, you get lots of morphisms everywhere, so you won't really be capturing any notion of composing solutions.