Timeline for What does an object of isomorphisms look like?

7 events

when toggle format	what		by	license	comment
Sep 10, 2013 at 23:56	comment	added	ct-novice		Does the object so constructed have a well-known name? Would I just call it the classifying object of isomorphisms?
Sep 10, 2013 at 23:55	vote	accept	ct-novice
Sep 9, 2013 at 23:23	comment	added	Colin McLarty		It is a good exercise to verify directly from the definition of exponentials that there is indeed an arrow $[A, B] \times [B, A] \to [A, A] \times [B, B]$ as described by Qiaochu.
Sep 9, 2013 at 23:07	history	edited	Qiaochu Yuan	CC BY-SA 3.0	deleted 9 characters in body
Sep 9, 2013 at 10:44	comment	added	Martin Brandenburg		Here is a more down-to-earth explanation of Qiaochu's answer: $\underline{\mathrm{Isom}}(A,B)$ should be a subobject of $\underline{\mathrm{Hom}}(A,B)$, namely of those morphisms which have an inverse. But since inverses are unique, we can also see it as the subobject of $\underline{\mathrm{Hom}}(A,B) \times \underline{\mathrm{Hom}}(B,A)$ of all $(f,g)$ such that $fg=1$ and $gf=1$. This is the preimage of $(1,1)$ under the map $\underline{\mathrm{Hom}}(A,B) \times \underline{\mathrm{Hom}}(B,A) \to \underline{\mathrm{Hom}}(A,A) \times \underline{\mathrm{Hom}}(B,B)$, $(f,g) \mapsto (gf,fg)$.
Sep 9, 2013 at 3:48	history	edited	Qiaochu Yuan	CC BY-SA 3.0	edited body
Sep 9, 2013 at 3:35	history	answered	Qiaochu Yuan	CC BY-SA 3.0