Demonstrating why category arrows are best thought of as arrows and not functions

Demonstration

Consider the category, [ilmath]\mathscr{C} [/ilmath], whose objects are finite sets and whose arrows, [ilmath]\xymatrix{A \ar[r]^f & B} [/ilmath] are functions:

• [ilmath]f:A\times B\rightarrow\mathbb{R} [/ilmath] with no imposed conditions.B

We define composition of arrows, given [ilmath]\xymatrix{A \ar[r]^f & B \ar[r]^g & C } [/ilmath], as:

• [ilmath]g\circ f:A\times C\rightarrow \mathbb{R} [/ilmath] as [ilmath]g\circ f:(a,c)\mapsto\sum\{f(a,y)g(y,c)\ \vert\ y\in B\}[/ilmath]

In words, [ilmath]g\circ f[/ilmath] maps [ilmath](a,c)[/ilmath] to the sum of (the products of the form [ilmath]f(a,y)g(y,c)[/ilmath] for any [ilmath]y\in B[/ilmath])

Claim: this is indeed a category.

Proof of claims

There's only one claim really.

Grade: C

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Discussion

You may think for a moment "I've seen this" and think Markov chains, or something to do with matrix multiplication, in truth it's just an example.

As required for a category, we can compose arrows, but for a change (even though the objects are sets, which isn't a change from what we usually deal with) composition of arrows isn't straight forward composition of functions.