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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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    I've voted to close because without a more specific definition of "problem", it is virtually impossible to answer this question. Please [edit] to include one or two concrete examples of morphisms that you want to capture; I'd be more than happy to remove my close vote.2015-03-11

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