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My background is in geometry, and I have a basic level of understanding of topological conjugacy: It defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring $f$ and $g$ to be equivalent if they are topologically conjugate ${\displaystyle g=h^{-1}\circ f\circ h}$ for some continous surjection $h$.

So topological conjugation is like a "change of coordinates". (And I saw orbits of $g$ are mapped to homeomorphic orbits of $f$ through the conjugation, but I am not totally sure what orbit means here.)

Question: What does conjugacy mean in algebra? I learned conjugacy classes of a group measure, in some way, the degree of its commutativity (if the conjugacy classes are just singletons, the group is abelian).

My broader question, how are these concepts of conjugacy linked, i.e. which broader mathematical idea is captured through conjugacy?

EDIT: I would like to thank you for excellent comments. They are extremely helpful! I am very grateful and would like to give upvote credits to these comments. If you post them as an answer, so I can give credit, that would be greatly appreciated!

R. Srivastava
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    Welcome to Mathematics Stack Exchange. Cf. this – J. W. Tanner Jul 21 '20 at 19:34
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    In linear algebra, a good case could be made that conjugacy by some matrix $B$ of $A$ corresponds to a change of basis in $A$ according to $B$. – Shaun Jul 21 '20 at 19:37
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    If $X$ is a topological space then the space of self maps $C(X, X)$ isn't necessarily a group, but it is a monoid. Not every element of a monoid is invertible, but we can still form conjugacy classes using the ones that are, i.e. say $a\sim b$ in the monoid $M$ iff there is an invertible $g\in M$ such that $a = g^{-1} b g$. So, the "topological conjugacy" in your question is in disguise a type of algebraic conjugacy for monoids, which directly generalizes the idea of conjugacy for groups. (This conjugacy for monoids analogously generalizes to categories.) – William Jul 21 '20 at 19:44
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    Thank you J.W. Tanner, Shaun, and William for really great answers! This is very helpful for me. William you make a great link for me between algebra and topology – R. Srivastava Jul 21 '20 at 19:54
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    Perhaps too secular, but I guess conjugacy is merely the closest relation to equality you can get for the building blocks $ab$ and $ba$, for every $a,b \in G$, where $G$ is a group. –  Jul 21 '20 at 20:02
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    To add an example from geometry, the Riemann curvature tensor measures the failure of covariant derivatives to commute. In the language of conjugacy, you can imagine flowing an infinitesimal amount in the horizontal direction, then flowing infinitesimally in the vertical direction, then flowing back in the horizontal direction (conjugation) -- if where you end up is not the same as just flowing vertically to begin with, you have curvature. This is related to ideas of parallel transport and holonomy -- when you go around and come back to where you started (conjugacy), things have changed! – Elchanan Solomon Jul 22 '20 at 01:31
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    @ElchananSolomon this is a great answer! It gives me a strong mental picture to which I can relate – R. Srivastava Jul 22 '20 at 03:50

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