A terminal object is isomorphic to an initial object if there is a morphism from T to I

up:: 027 MOC Category Theory

We have that every Initial Object has a unique morphism coming out of it towards any other object in a Category , and that every Terminal Object has a unique object coming into it. Thus, there is only one morphism from to ; call it .

In the case where ¹, we can see that

Therefore, we have that is the inverse of ─ therefore, is an isomorphism.

Corollaries

When there is a morphism from a terminal object to an initial object, we have that they are isomorphic. In this case, they will be isomorphic to a Zero Object. Thus, there can only be one zero object (up to isomorphism) in a category.

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References

• KASHIWARA, Masaki; SCHAPIRA, Pierre, Categories and Sheaves, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.

Footnotes

1. Note that this doesn’t make stop being a terminal object; it is an additional property that it possesses, aside from being terminal. ↩