Talk:Free object

Master

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

This suggestion does not have any content bearing on the topic.

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

Correction

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)

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

Free universal algebras

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

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

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

"Free universal algebras" section

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

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)

Examples

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

The notation $F_{2}$ 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 $F_{2},$ 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)

Bug in the Definition?

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)

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)

Upgrading to tikz diagram

@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)

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)

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)