Talk:Free object

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"It is probably better to master some special case such as free groups first."

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I agree. I'll cut that.--345Kai (talk) 22:59, 8 May 2009 (UTC)[reply]

The first section, the introduction, glosses a point: the set of words is the appropriate starting point only for associative algebras, otherwise, it needs to be the magma of binary trees. I will clarify/fix this later; I am going to an Easter party now. linas 18:03, 8 April 2007 (UTC)[reply]

Have a good time. I've added a note covering this issue. Charles Matthews 18:29, 8 April 2007 (UTC)[reply]

An example of a free algebra. Let ${\displaystyle X}$ be an arbitrary set and consider the vector spaces over the field ${\displaystyle K}$. ${\displaystyle (F(X),h)}$ is the freely generated algebra where

${\displaystyle h(x):X\longrightarrow K}$. ${\displaystyle h(x)(y)=1}$ if ${\displaystyle x=y}$, ${\displaystyle 0}$ otherwise.

${\displaystyle F(X)=\{\sigma :X\longrightarrow K|\{x\in X|\sigma (x)\neq 0\}{\mbox{ finite }}\}}$

Should the last part be "there exists a unique homomorphism ${\displaystyle \sigma :\mathbf {A} \longrightarrow \mathbf {B} }$ such that ${\displaystyle \sigma \psi =\tau }$."?

Otherwise, I don't know where the sigma came from or what it's supposed to mean.--kundor (talk) 14:18, 8 July 2008 (UTC)[reply]

The main example of a free group is very confusing with respect to the formal definition. What corresponds to ${\displaystyle X}$, ${\displaystyle i}$, ${\displaystyle A}$ and ${\displaystyle F}$ in the example? Why does ${\displaystyle F_{2}}$ have a subscript 2? — Preceding unsigned comment added by Eposse (talk • contribs) 16:26, 31 May 2020 (UTC)[reply]

The notation ${\displaystyle F_{2}}$ refers to the Free group with 2 generators a and b. The set X is the set of generators, here {a, b}. In section "Definition", A denotes the free object, here ${\displaystyle F_{2},}$ anf i is the injection of X in A, which, here, maps a and b viewed as elements of the set X to themselve viewed as elements of the free group.

I agree that the article is badly written, in a pedantic style (systematic use of technical terms that are not mastered by most readers, the same thing can be comprehensibly said with few more words). D.Lazard (talk) 17:38, 31 May 2020 (UTC)[reply]

What is the parameter $A$ in the term $F(A)$ in the definition? I guess it should be rather $F(X)$... So all $X$ or all $A$ 2A01:598:D007:4610:74C9:8F80:2E52:CA5F (talk) 15:04, 1 July 2022 (UTC)[reply]

A is the free object to be defined. However, I'll clarify that X is a set and A is an object of C. D.Lazard (talk) 16:11, 1 July 2022 (UTC)[reply]

@D.Lazard I noticed that you reverted a few changes. Could you explain a bit? For example, I thought it made sense to upgrade the commutative diagram to tikz for consistency with universal property. Davidvandebunte (talk) 17:37, 1 April 2023 (UTC)[reply]

Sorry, I did not see that you did edit yestersay. I intended to revert the edits of today, concerning nlab references. I have restored your edits that are not for linking nlab. The removal of these links is explained in

WP:USERGENERATED. D.Lazard (talk) 17:57, 1 April 2023 (UTC)[reply

]

OK thanks, that makes sense. I've generally made exception for nlab because it is usually edited by authoritative authors in category theory like John Baez, and I'd seen it used as a reference in other articles (see Adjoint functors). But, I can see your point. Perhaps this could go under an "External links" section as in Category theory? Davidvandebunte (talk) 18:19, 1 April 2023 (UTC)[reply]