Thin groups and also charts (isomorphism of groups )

I attempt to find out the concept of group from The Joy of Cats. I obtained piled with the first workout (3A a ).

If we have a straightforward chart with one vertex and also 2 nodes all we understand is that in group there is 2 things and also one homomorphism (close to identifications ). Therefore :

\[C = (\{\{x, y\}, \{1, 2\}\}, \{f, id\}, id, \cdot), f = \{(x, 1), (y, 2)\}\]

and also

\[C = (\{\{x\}, \{1\}, \{f, id\}, id, \cdot), f = \{(x, 1)\}\]

Should have the very same chart yet are they isomorphic?

Keep in mind concerning symbols : I deal with group as quadruple $(O, H, id, \cdot)$ where $O$ is class of things (below is set ), $H$ is class of homomorphisms (in instance set ). $id$ is identification and also $\cdot$ is make-up of homomorphisms.

2019-05-07 12:41:16
Source Share
Answers: 1

It is a little bit difficult to comprehend your symbols. If recognize appropriately, the isomorphism is understood by the functor $F$ which sends out $\{x, y\}$ to $\{ x \}$ and also $\{ 1, 2 \}$ to $\{ 1 \}$ (on things) and also send the one-of-a-kind non unimportant morphism $f$ to $f$.

2019-05-09 06:32:44