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I have read at many places that the best way to study about a object $X$ of some category,the "best" way is to study the morphisms from $X \to X$.My question is:

What can be learn about a object $X$ by studying morphisms from $X \to X$ ?

I am mainly interested in the following categories:

$1$ Category of Groups where morphisms are homomorphism

$2$ Category of Topological Spaces where morphism are Continuous maps

$3$ Category of Vector Spaces where morphisms are Linear maps

$4$ Category of Measurable spaces where morphisms are measurable maps.

$5$ Category of Lie Algebras where morphisms are Lie maps.

P.S: I don't have any deep idea about category theory,i know the basics of category theory.

Arpit Kansal
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    This is sort of a strange claim in general. Often this is a very complicated object (say for topological spaces) so not much can be said; in these settings it's better to think about morphisms from $X$ into or out of some simpler object. Things get considerably better if the category you're working in is additive; then endomorphisms form a ring, and for example, idempotents in that ring correspond to direct summands. Of your examples this only applies to 3. – Qiaochu Yuan Aug 19 '16 at 06:47
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    Dear @QiaochuYuan: I see..suppose we study the morphisms from $X $ into or out of some simpler object.Then can we recover the object completely? Also, what will be those simpler object for instance for category of measurable spaces? – Arpit Kansal Aug 19 '16 at 07:17
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    In many cases, yes. The claim that if you take all other possible objects then you can recover $X$ is essentially the Yoneda lemma. it is usually possible to do better than this. For example, to recover a group $G$ together with its group operation it suffices to consider morphisms from finitely generated free groups into $G$. – Qiaochu Yuan Aug 19 '16 at 07:47
  • With regard to your original question see http://math.stackexchange.com/questions/801585/can-finite-non-isomorphic-groups-of-the-same-order-have-isomorphic-endomorphism – Nex Aug 19 '16 at 07:50
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    You've really read in many places that "the best way to study about a object X of some category is to study the morphisms X→X"? Can you give one example of a place you've read that? – Omar Antolín-Camarena Aug 19 '16 at 22:57
  • Dear @OmarAntolín-Camarena: I don't know whether the statement is given in any book or not but i have read this in some questions of MSE. For example,http://math.stackexchange.com/questions/120147/why-do-we-look-at-morphisms. – Arpit Kansal Aug 21 '16 at 10:03
  • If you read that question carefully you'll see that it doesn't say that the best way to study an individual object $X$ is to study the endomorphism $X\to X$, but rather that to study a class of objects one should study the morphisms between pairs of them. That is, it doesn't say "the best way to study the monster group $M$ is through it's monoid of endomorphism" but rather "to study the theory of groups it is best to study group homomorphisms". I doubt you'll actually find the endomorphism statement anywhere. – Omar Antolín-Camarena Aug 21 '16 at 15:56
  • Dear @OmarAntolín-Camarena.You're right.I think its too much to expect to recover the object just by studying morphisms $X\to X$. By your comment and other several comments/answer i think the conclusion is that the "best" way is to think about morphisms from $X$ into or out of simpler object instead of mor$(X,X)$. Could you please tell me something more about the category of measurable spaces ?Like what are those "simpler" objects we should concentrate on?Thank you. – Arpit Kansal Aug 21 '16 at 16:08
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    In measure theory people usually look at functions on a measure space, that is morphisms $X \to \mathbb{R}$ where $\mathbb{R}$ is equipped with the $\sigma$-algebra of Boreal sets. – Omar Antolín-Camarena Aug 22 '16 at 03:52

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In some sense, this is a very broad question.

Even prior to category theory, some people like to study objects by studying groups acting on those objects.

For example, the Euclidean group $E(2)$ acts on the Euclidean plane.

It has a normal subgroup $SE(2)$ such that $E(2) / SE(2) \cong \mathbb{Z} / 2 \mathbb{Z}$; this tells us that the Euclidean plane has a notion of orientation. Note that this fact arises simply from the group structure of $E(2)$, without any prior knowledge of the Euclidean plane.

$E(2)$ has a normal subgroup $T(2)$, the "translation group", and $E(2) / T(2) \cong O(2)$, the orthogonal group. This tells us:

  • The cosets of $O(2)$ are some sort of 'geometric' object that is unchanged by orthogonal transformations
  • Between any two such objects, there is a unique element of $T(2)$ that "translates" from one to the other

In other words, the Euclidean plane is made out of points in the familiar way.

Now, I don't think I'd say category theory encourages this at all, aside from providing tools for more doing more sophisticated analyses.

The thing category theory encourages is thinking of the morphisms $Y \to X$ for any $Y$ (or maybe for any $Y$ from a particularly nice subset of objects) as being a good substitute for the notion of "element of $X$" — in fact, it's arguably a better notion of "element" than the ordinary notion of element when one exists.