Talk:Free object

"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)

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)

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

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

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

Should the last part be

"there exists a unique homomorphism $\backslash sigma\; :\backslash mathbf\{A\}\; \backslash longrightarrow\; \backslash mathbf\{B\}$ such that $\backslash sigma\; \backslash psi\; =\; \backslash tau$."?

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)

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 {{mvar|a}} and {{mvar|b}}. The set {{mvar|X}} is the set of generators, here {{math|{{mset|a, b}}}}. In section "Definition", {{mvar|A}} denotes the free object, here $F\_2,$ anf {{mvar|i}} is the injection of {{mvar|X}} in {{mvar|A}}, which, here, maps {{mvar|a}} and {{mvar|b}} viewed as elements of the set {{mvar|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)

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)

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

@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)