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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[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]

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)[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]