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# Exercise 31 - Chapter 3

edited June 2018

Recall Examples 3.12 and 3.17. A monoid in which every morphism is an isomorphism is known as a group.

1) Is the monoid in a group?

2) What about the monoid in and ?

2) $$\mathcal{C}$$: Yes, $$s^{-1}=s,\mathrm{id}_z^{-1}=\mathrm{id}_z$$. $$\mathcal{D}$$: No, there are no inverses for non-identity morphisms.
Comment Source:1) No, there are no inverses for non-identity morphisms. 2) \$$\mathcal{C}\$$: Yes, \$$s^{-1}=s,\mathrm{id}_z^{-1}=\mathrm{id}_z\$$. \$$\mathcal{D}\$$: No, there are no inverses for non-identity morphisms.