Talk:Initial and terminal objects: Difference between revisions

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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 [[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)

== zero group ==

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

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

== 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. [[Special:Contributions/74.71.239.188|74.71.239.188]] ([[User talk:74.71.239.188|talk]]) 18:22, 27 April 2010 (UTC)

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. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/158.121.231.73|158.121.231.73]] ([[User talk:158.121.231.73|talk]]) 22:17, 27 December 2011 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

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

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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? ==

0The doesn'tfigure injectseems intoto Z.show an object {0} mustthat mapis to 0terminal, but asnot ainitial unital(no homomorphism,arrows from {0 must also map} to 1. Soany inof the ringsother withelements unity,of 0 can't be homomorphically mapped into ZA). <span!-- style="font-sizeTemplate:Unsigned smaller;"IP --><small class="autosigned">— &nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/15898.121207.231232.73114|15898.121207.231232.73114]] ([[User talk:15898.121207.231232.73114#top|talk]]) 22:1725, 2718 DecemberFebruary 20112018 (UTC)</span><!-- Template:Unsigned IP --small> <!--Autosigned by SineBot-->