Talk:Initial and terminal objects

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Rename to universal object?

What do people think about renaming this page to universal object? The primary advantage is that this name treats initial and terminal objects on equal footing. After all, this article is just as much about terminal objects (and zero objects) as it is initial objects. The primary disadvantage is that the name universal object is not nearly as common as initial or terminal object. It is, however, used in this sense—see, for example, Lang's Algebra or Hungerford's Algebra. Numerous other instances can be found (excepting, notably, Mac Lane's monograph).

I would still suggest we use the terminology initial object and terminal object in the article itself. Which of these should be universal and which should be couniversal is surely going to vary from author to author. Hungerford, for example, calls initial objects universal and terminal ones couniversal, but this is at odds with the usage of limits and colimits. Lang calls initial objects universal repelling and terminal objects universally attracting which is slightly more descriptive. -- Fropuff (talk) 19:47, 15 January 2008 (UTC)[reply]

zero group

why is the empty set initial in set, but not in group?

Because there is no empty group. Algebraist 01:41, 17 December 2009 (UTC)[reply]

something foul about example

Z is initial in unital rings, unit preserving homomorphisms, and 0=1 is terminal. I guess 0 doesn't inject into Z, since that would give 2 morphisms Z -> Z, (and in a sense not be unit preserving). Is this right? It seems weird. 74.71.239.188 (talk) 18:22, 27 April 2010 (UTC)[reply]

0 doesn't inject into Z. 0 must map to 0, but as a unital homomorphism, 0 must also map to 1. So in the rings with unity, 0 can't be homomorphically mapped into Z. — Preceding unsigned comment added by 158.121.231.73 (talk) 22:17, 27 December 2011 (UTC)[reply]

Shall we not avoid to talk about "Category of semi-groups" or "Category of non-empty Sets"

The formal definition of a category implies the existence of an identity arrow for each object. Because semi-groups do not have an identity element, I guess we should not call them "category of semigroups". Shall we not call them "questionable category of semigroups" or something in that spirit? — Preceding unsigned comment added by 178.197.234.69 (talk) 14:12, 24 October 2016 (UTC)[reply]

In the case of categories whose objects are sets or which have an underlying set, the identity arrow is the identity mapping from the object to itself. This is the case here. For example, in the category of non-empty sets, the objects are sets and the arrows are mappings from a set to another (or to the same) set. This has to not be confused with the category that can be associated to a specific monoid, which has only one object and whose arrows are the elements of the monoid. Contrarily to preceding examples the category associated to a monoid has only one identity element, while the category of sets (or of monoids) has many identity arrows (one for each set or monoid). D.Lazard (talk) 16:28, 24 October 2016 (UTC)[reply]