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Definition 1.2 If $C$ is a concrete category, $F \in Obj(C)$, and $X$ is a set, and if $\phi : X \to \sigma(F)$ is a one-to-one function, then $F$ is called free on $X$ if and only if for every $A \in Obj(C)$ and set map $\psi : X \to \sigma(A)$, there exists a unique morphism $f \in Mor(F, A)$ such that $f \circ \phi = \psi$ as set maps from $X$ to $\sigma(A)$.

$\sigma$ is the map that comes together with the concrete category $C$, the map that makes $C$ into a concrete category.

Thanks. It's just that this definition is so abstract.

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  • $\begingroup$ Do you understand particular examples, when $C$ is, say, $\mathbf{Grp}$ and $\mathbf{CAlg}_k$. $\endgroup$ – Alex Youcis Nov 5 '13 at 0:10
  • $\begingroup$ I might understand Grp. $\endgroup$ – BananaCats Category Theory App Nov 5 '13 at 0:11
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    $\begingroup$ I mean, they're basically the objects that everything is a "quotient" of. Their use is, in some sense, undeniable. How often in field theory did you say "Ok, let $L/K$ be an extension, and let $\alpha\in L$. We know there is a $K$-algebra map $K[x]\to L$ with $x\mapsto \alpha$..." You are using PRECISELY that $K[x]$ is the "free commutative $K$-algebra on one generator". $\endgroup$ – Alex Youcis Nov 5 '13 at 0:14
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Consider the category $\mathbf{\mathbb{K}\text{-}Vect}$ of $\mathbb{K}$-vector spaces. Let $V$ be a vector space, $B$ its basis and $\phi\colon B\to\sigma(V)$ the obvious embedding. Then $V$ is free on $B$, because the condition for freeness translates exactly to the property, that any linear map between vector spaces is completely determined by its action on the basis vectors.

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    $\begingroup$ And next check the meaning Abelian groups (or more generally R-modules) and see that not every object is free. $\endgroup$ – Carsten S Nov 5 '13 at 9:38
  • $\begingroup$ Thanks! This is an example I can understand! Apparently you don't have to understand all of Math like that other answerer said, before learning Category theory. $\endgroup$ – BananaCats Category Theory App Nov 6 '13 at 15:00
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A way to think about it : a free object on the set $X$ respects the universal property that $F(X)$ would respect if the forgetful functor $U \colon C \to \mathbf{Sets}$ had a left adjoint $F$.

So now, just think about what is a left adjoint of $U$ : it is a functor $F \colon \mathbf{Sets} \to C$ which comes with a (natural) bijection for all object $Y$ of $C$ $$ \hom_{C}(F(X), Y) \simeq \hom_{\mathbf{Sets}}(X,U(Y)).$$ It means that defining a map from a free object on the set $X$ to the object $Y$ is the same as defining a map form $X$ to the underlying set of $Y$. Which is precisely what you do with the usual free object you (should) know : free groups, free modules, etc.

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You shouldn't learn category theory now if you don't know immediately at least three examples for free constructions. Or you should choose another textbook, because any good textbook mentions basic examples from which one then abstracts.

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