Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Confusion on the Nature of Isomorphisms in HomSet

Jacob Zelko (Jun 25 2022 at 02:49):

[Beginner Question]: I attempted to answer the question:

Find a set A such that for any set X, there is an isomorphism of sets $X \cong Hom_{Set}(A, X)$.

Which came from Category Theory for the Sciences by Spivak.
I tried answering the question as follows:

Going back to the definition of an isomorphism, I knew that there must exist a bijective relationship between sets $A$ and $X$ for any set $X$.
My answer to this was that $A$ must be any set that is equal to the number of elements in $X$.
Upon looking at the solution to this problem, I was surprised to realize that my understanding of isomorphisms and HomSets appear wrong:

Let $A = \{\pi\}$ Then to point each element of $A$ to an element of $X$, one must simply point $\pi$ to an element of $X$. The set of ways to do that can be put in one-to-one correspondence with the set of elements of $X$. For example, if $X = \{1, 2, 3\}$, then $\pi \mapsto 3$ is a function $A \rightarrow X$ representing the element $3 \in X$.

What I expected in the latter example is that $A$ would have 3 elements like when $X = \{1, 2, 3\}$ to represent satisfy a complete one to one mapping between each set.
However, this does not appear to be the case and now I am quite confused as to why we can get away with saying that $A$ can have only one element that could map to only one element of a set.
Why is this the case?

UPDATE: While writing out my question, I realized that an isomorphism can be both bijective and injective only.
Therefore, I realized I do understand isomorphisms but I failed to recognize the injective nature of isomorphisms as well making the above solution completely valid.
Just to check, did I correct myself properly here or am I still wrong?

John Baez (Jun 25 2022 at 04:36):

I think you have it right now. Isomorphisms between sets are the same as bijections. Two sets are isomorphic iff they have the same number of elements. We can use this to figure things out:

The number of functions from a set $A$ to a set $X$ is $|X|^{|A|}$, where $|A|$ is the number of elements of $|A|$ and $|X|$ is the number of elements of $X$. For the set of functions from $A$ to $X$ to be isomorphic to $X$, we need their number of elements to be equal:

$|X|^{|A|} = |X|$

This is only true for all $X$ if $|A| = 1$. So, we need $A$ to have 1 element.

And you checked that taking $A$ to have one element works! Then we get a bijection sending any function $f: A \to X$ to $f(a)$, where $a$ is the one element of $A$.

Jacob Zelko (Jun 25 2022 at 21:45):

John, Thank you! this was a great response and I am on the same page as you now. What I think got me on the wrong track is that I thought that each set must be bijective to another set and not that each element between sets must be bijective. though I did recognize this fact $|X|^{|A|} = |X|$ I did not have the correct foresight to set up that equality as I was thrown off by the phrasing "isomorphism of sets" which made me think to just jump to trying to write out possible sets. I should have written out what I knew about the state ments in the question first. Thank you so much for working with me through this! I feel like this cleared up my confusion! Have a great day @John Baez !

John Baez (Jun 25 2022 at 21:48):

Great!

John Baez (Jun 25 2022 at 21:49):

"Isomorphism in Set" is just another name for "bijection"; the only advantage of the former terminology is that isomorphisms make sense in any category, and once you get used to category theory you may get tired of separate names for isomorphisms in different categories: bijections, homeomorphisms, diffeomorphisms, etc.

Jacob Zelko (Jun 25 2022 at 21:50):

Ope! @John Baez . Should I not mark problems solved on this Zulip? my bad if I broke a rule! :grimacing:

John Baez (Jun 25 2022 at 21:52):

It's not a rule, but we never do it, and when I tried to answer your last comment Zulip complained to me that I was talking about a resolved problem.

John Baez (Jun 25 2022 at 21:52):

I think that "resolved" stuff is more for people who use Zulip in computer programming. In math nothing is ever completely resolved; we talk forever.

Jacob Zelko (Jun 25 2022 at 21:53):

John Baez said:

"Isomorphism in Set" is just another name for "bijection"; the only advantage of the former terminology is that isomorphisms make sense in any category, and once you get used to category theory you may get tired of separate names for isomorphisms in different categories: bijections, homeomorphisms, diffeomorphisms, etc.

I can definitely see that being cumbersome to worry about! I will focus on keeping this more straight in my mind when thinking on categories. also weird about the Zulip! will not do that going forward.

Jacob Zelko (Jun 25 2022 at 21:55):

In math nothing is ever completely resolved; we talk forever.

Haha, Love that! I can see that and somewhere, Kuhn smiles.