Notation for set of maps.

The set of all homomorphisms between two spaces $$X$$ and $$Y$$ is denoted as $$\text{Hom}(X,Y)$$, the set of endomorphisms of a space $$X$$ is denoted as $$\text{End}(X)$$,... these are standard notations, but what about the set of all maps between two spaces $$X$$ and $$Y$$ (without any extra structure) Are $$\text{Map}(X,Y)$$ or $$\text{Fun}(X,Y)$$ standard notations?

This might be an stupid question, but I want to make sure I am using the right notation, or at least the most usual notation for this spaces. Thanks for your help.

• Notations are defined. If you feel weird when using these notations, please define them first. I I have no impression about a notation for the function space between $X$ and $Y$. Jan 24, 2022 at 16:59
• you can use $Y^X$ Jan 24, 2022 at 17:01
• @xfireskyx Yeah, I know, but when possible it is better to use standard notations. For example, I could define the derivative of $f$ as $\hat{f}$, but it is not standard and no one will understand it. Jan 24, 2022 at 17:01
• It all depends on context. Those are certainly used and are more common to see on the algebra side of things. From a combinatorics side of things you might see $Y^X$ used to denote the set of functions from $X$ to $Y$ instead which assists with counting as one would have $|Y^X|=|Y|^{|X|}$ Jan 24, 2022 at 17:01
• @JMoravitz Right, thanks for your answer. Jan 24, 2022 at 17:06

Common ways to denote the set of all functions from a set $$X$$ to a set $$Y$$ that I have encountered (some of which have already been mentioned) are $$\mathcal{F}(X;Y)$$, $$\text{Map}(X;Y)$$ and $$Y^X$$. Then you can define $$\mathcal{F}(X):=\mathcal{F}(X;X)$$ etc. I don't think there is a 'standard' way of presenting such functions that has anywhere near a consensus. I would personally use whichever notation feels most natural, depending on the context.
The $$Y^X$$ notation comes from the fact that you are choosing one value of $$Y$$ for each value of $$X$$, so you can kind of think of it is an '$$X$$-fold Cartesian product' in this sense i.e. $$X$$ copies of the set $$Y$$.
As mentioned in the comments, the notation $$Y^X$$ is standard for the set of maps from $$X$$ to $$Y$$. Moreover, there is no other common notation for this.