# Making a Family of Problems into a Category and defining the Morphisms

This question relates to my previous question found here: Defining Category of Problems

Let $\left\{\Pi_i \right\}_{i \in I}$ be a family of problems. Let the solution $u_i$ of $\Pi_i$ lie in some space $X_i$. I want to make this family into a category in order to study decomposition of problems into products of smaller ones. I want to define the morphisms of this category in a manner that, the existence of a morphism $f_{j}^i : \Pi_i \rightarrow \Pi_j$ is equivalent to saying that solving $\Pi_i$ implies solving $\Pi_j$.

(from here and onwards re-edited)

I define a morphism $f_{j}^i : \Pi_i \rightarrow \Pi_j$ to be a map $f: X_i \rightarrow X_j$ such that $f(u_i) = u_j$ and $f$ does not depend on $u_j$.

Question: does this definition capture the interpretation of morphisms that i want? I.e. let $f_{j}^i : \Pi_i \rightarrow \Pi_j$ be a morphism. Does it follow that solving $\Pi_i$ implies solving $\Pi_j$?

thanks :-)

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You can always decide which diagrams you want to use. These diagrams decide which morphisms are available and which are not. – tp1 Nov 17 '11 at 23:50
Is the problem of defining the morphisms an object in the Category of problems? :-) – Asaf Karagila Nov 17 '11 at 23:55
@tp1: I think i see your point:) Could you please give me a little bit more detail? So what you are saying is that given the family of problems, the arrows are defined automatically. Right? – Manos Nov 17 '11 at 23:55
Interesting thoughts! but what is your formal definition of "problem"? – Bruno Joyal Nov 18 '11 at 0:00
@Bruno: Thanks. I don't have a formal definition for what a problem is. I am thinking of a well-defined mathematical problem. Maybe we can think of it as a map from the space of parameters of the problem to the solution space, e.g. solve $\alpha x = \beta \in \mathbb{R}$, i can think of it as map $\mathbb{R}^2 \rightarrow \mathbb{R}$. – Manos Nov 18 '11 at 0:06