In Chapter 1.7 of Spanier's Algebraic Topology, Spanier writes:

If $A$ and $B$ are in the same component of a groupoid, the collection of isomorphisms $\{h_{f}\mid f:A\rightarrow B\}$ is a conjugacy class of isomorphisms $hom(A,A)\rightarrow hom(B,B)$.

By ${h_{f}}$ he means the group isomomorphism $hom(A,A)\rightarrow hom(B,B)$ which maps each morphism $g:A\rightarrow A$ to $f\circ g\circ f^{-1}$. I do not understand what he means by "conjugacy class of isomorphisms $hom(A,A)\rightarrow hom(B,B)$". From what I understand, a conjugacy class is a set of the form $\left\{ bab^{-1}\mid b\in G\right\}$ for some $a\in G$ where $G$ is some group. I get that for each object $A$ in the groupoid, $hom(A,A)$ is a group under composition but what I don't get is how the isomorphisms $hom(A,A)\rightarrow hom(B,B)$ are elements of a group related to the groups on the hom sets, so the statement "conjugacy class of isomorphisms $hom(A,A)\rightarrow hom(B,B)$" makes absolutely no sense to me.

Note this question has been asked once before but I didn't feel that the reply was satisfactory.

  • $\begingroup$ A link to the previous version of this question, please? $\endgroup$ Dec 4, 2016 at 22:19
  • $\begingroup$ And could you clarify why you feel the reply was unsatisfactory? (I presume the previous version was this question.) $\endgroup$ Dec 4, 2016 at 23:04
  • $\begingroup$ Yes it was that question, and I thought it was unsatisfactory because the person who replied didn't actually answer the question. $\endgroup$
    – Eigenfield
    Dec 5, 2016 at 5:49


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