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)