Can someone please give an example of a function between two categories that is an object function, arrow function and preserves the identity arrow(s) but fails to preserve the composition property.
Thank you!
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An easy way to produce a counterexample is to look at monoids (which are the same thing as categories with one object) : it suffices to find a function between two monoids that preserve the identity but not the composition. For example, you can take a non-abelian group $G$, and the function $g\mapsto g^2$.
For a similar example, but with a category with more objects : take the category $\mathbf{Mat}_{\Bbb R}$ whose objects are the natural numbers and arrows $m\to n$ are matrices in $\mathbb{R}^{n\times m}$, with composition defined by the matrix product. Then you can define functions $\mathbb{R}^{n\times m}\to \mathbb{R}^{n\times m}:A\mapsto AA^tA$ for all $n,m$. Together with the identity function on $\mathbb{N}$, this respects domains, codomains and identities, but not composition, because $AA^tABB^tB\neq ABB^tA^tAB$ in general.
Arnaud D.
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Thank you. This helped. – Temari Jun 10 '18 at 09:43
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For another abstract example, consider the preordered set $C:=(\{0,1,2\}, \le) $ as a category (i.e. besides the 3 identity morphisms, it has arrows $a:0\to1,\ b:1\to 2,\ c:0\to2$).
Let $D$ be obtained by adding one more morphism $d:0\to 2$ to $C$ (but keeping $b\circ a=c$).
Now define $F:C\to D$ by being the identity on objects and identity morphisms, and taking $a\mapsto a, \ b\mapsto b\ $ and $\ c\mapsto d$.
Berci
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