Talk:Initial and terminal objects: Difference between revisions

Revision as of 19:47, 15 January 2008

Latest revision as of 01:22, 9 March 2024

(17 intermediate revisions by 13 users not shown)

Line 1:

{{WikiProject banner shell|class=Start|vital=yes|1=

{{maths rating|class=Stub|priority=Mid|field=foundations}}

{{WikiProject Mathematics|priority=Mid}}

}}

{{archives|auto=short|search=yes}}

==Rename to universal object?==

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

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).

The formal definition of a category implies the existence of an identity arrow for each object.

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 [[limit (category theory)|limits and colimits]]. Lang calls initial objects ''universal repelling'' and terminal objects ''universally attracting'' which is slightly more descriptive. -- [[User:Fropuff|Fropuff]] ([[User talk:Fropuff|talk]]) 19:47, 15 January 2008 (UTC)

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? <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/178.197.234.69|178.197.234.69]] ([[User talk:178.197.234.69#top|talk]]) 14:12, 24 October 2016 (UTC)</small> <!--Autosigned by SineBot-->

: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). [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:28, 24 October 2016 (UTC)

== Figure is wrong? ==

The figure seems to show an object {0} that is terminal, but not initial (no arrows from {0} to any of the other elements of A). <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/98.207.232.114|98.207.232.114]] ([[User talk:98.207.232.114#top|talk]]) 22:25, 18 February 2018 (UTC)</small> <!--Autosigned by SineBot-->