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Post Closed as "Needs more focus" by R. van Dobben de Bruyn, Wolfgang, Andy Putman, Peter LeFanu Lumsdaine, Gro-Tsen
Post Made Community Wiki by Todd Trimble
added 5 characters in body; edited title
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General principles which lead to good questions in many concrete situations

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such principles. Below is my own list.

  1. Given a Banach space, one may ask what is its dual. In many concrete situations this question has interesting and important answers. (For example the dual space of the space of continuous functions on a compact space is the space of measures on it.)

  2. Given a functor between two categories, one may ask whether is has right/ left adjoint. One may also ask whether it has right/ left derived functors. (For example the simple operation of push-forward of sheaves of abelian groups has left adjoint - pull back- which is somewhat less obvious.)

  3. Given a metric space, one may ask what is its completion. (Concrete example which I like: consider the set of isometry classes of $n$-dimensional closed Riemannian manifolds of diameter at most $D$ and sectional curvature at least $\kappa$ equipped with the Gromov-Hausdorff metric. Points of its completion are compact metric spaces which are so called Alexandrov spaces with curvature bounded below.)

General principles which lead to good questions in concrete situations

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in concrete situations. I would like to have a list of such principles. Below is my own list.

  1. Given a Banach space, one may ask what is its dual. In many concrete situations this question has interesting and important answers. (For example the dual space of the space of continuous functions on a compact space is the space of measures on it.)

  2. Given a functor between two categories, one may ask whether is has right/ left adjoint. One may also ask whether it has right/ left derived functors. (For example the simple operation of push-forward of sheaves of abelian groups has left adjoint - pull back- which is somewhat less obvious.)

  3. Given a metric space, one may ask what is its completion. (Concrete example which I like: consider the set of isometry classes of $n$-dimensional closed Riemannian manifolds of diameter at most $D$ and sectional curvature at least $\kappa$ equipped with the Gromov-Hausdorff metric. Points of its completion are compact metric spaces which are so called Alexandrov spaces with curvature bounded below.)

General principles which lead to good questions in many concrete situations

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such principles. Below is my own list.

  1. Given a Banach space, one may ask what is its dual. In many concrete situations this question has interesting and important answers. (For example the dual space of the space of continuous functions on a compact space is the space of measures on it.)

  2. Given a functor between two categories, one may ask whether is has right/ left adjoint. One may also ask whether it has right/ left derived functors. (For example the simple operation of push-forward of sheaves of abelian groups has left adjoint - pull back- which is somewhat less obvious.)

  3. Given a metric space, one may ask what is its completion. (Concrete example which I like: consider the set of isometry classes of $n$-dimensional closed Riemannian manifolds of diameter at most $D$ and sectional curvature at least $\kappa$ equipped with the Gromov-Hausdorff metric. Points of its completion are compact metric spaces which are so called Alexandrov spaces with curvature bounded below.)

Source Link
asv
asv
  • 21.8k
  • 6
  • 54
  • 121

General principles which lead to good questions in concrete situations

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in concrete situations. I would like to have a list of such principles. Below is my own list.

  1. Given a Banach space, one may ask what is its dual. In many concrete situations this question has interesting and important answers. (For example the dual space of the space of continuous functions on a compact space is the space of measures on it.)

  2. Given a functor between two categories, one may ask whether is has right/ left adjoint. One may also ask whether it has right/ left derived functors. (For example the simple operation of push-forward of sheaves of abelian groups has left adjoint - pull back- which is somewhat less obvious.)

  3. Given a metric space, one may ask what is its completion. (Concrete example which I like: consider the set of isometry classes of $n$-dimensional closed Riemannian manifolds of diameter at most $D$ and sectional curvature at least $\kappa$ equipped with the Gromov-Hausdorff metric. Points of its completion are compact metric spaces which are so called Alexandrov spaces with curvature bounded below.)