What is the mathematical understanding behind what physicists call a gauge fixing?

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I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an overlapping patch of the base space of the bundle. Said differently a gauge transform is a change of section in overlapping spaces. (Comments about that are welcome :-)

I'm wondering what is the meaning of a gauge fixing in term of fiber bundle model. Is that a reduction/constraint of the group transformation between sections for instance ?

Thanks for any comment which can improve this question.