Talk:Comma category

message: So I just expanded this quite a bit - some of the previous phrasing has been incidentally squelched, but everything that was there before is still there now. I'm pretty sure that the new stuff is accurate, but I'm not too happy with the notation: there are far too many different symbols. It could also use a commutative diagram for the most general case. The material on adjoints is currently minimal; I've tried to say just enough to show "look, there's a connection!" without actually going into any details, but perhaps further details are needed. --AlexG 00:44, 19 Jul 2004 (UTC)

message: I'm new to category theory, so I won't make changes myself here. But I want to suggest the following:
# There's no definition of the identity morphism of each object in the comma category.
# By the given definition of the morphisms, it is not specified but I suppose it must be obvious that the morphisms between two objects are taken uniquely. For example: if there are objects (a,b,f1) (a',b',f1') (a,b,f2) (a',b',f2') in the comma category, then there could be g:a->a' h:b->b' s.t. (g,h) is a morphism both in hom((a,b,f1), (a',b',f1')) and hom((a,b,f2), (a',b',f2')), but these are two separate morphism with the same notation, isn't it so?
--Itaj Sherman (talk) 23:28, 15 February 2008 (UTC)
replies: text: The identity morphism for (a, b, f) is (ida, idb).  You are right that the notation for morphisms is a bit ambiguous.  Normally one thinks of the morphism as being given with its domain and codomain, but sometimes writing a map properly — (g, h) : (a, b, f) → (a’, b’, f’) — requires too much notation.  I found a reference for the identity morphism in Adámek–Herrlich–Strecker, which does not comment on the ambiguity.  It's possible that Mac Lane talks about it, but I don't have my copy handy at the moment. Michael Slone (talk) 01:29, 16 February 2008 (UTC)
replies: text: Well, I think it should be at least noted that the meaning of the definition is to be exclusive per domain and codomain, whether the books state it or not, this is what they mean. It would be helpful for people who are new to category theory - I was confused and had to go through all the definitions again to understand what was meant there. --Itaj Sherman (talk) 00:00, 23 February 2008 (UTC)
replies: text: There really shouldn't be any confusion here. Two morphisms with different domains (or codomains) are never equal. This isn't special to comma categories. I agree that the language used in the article is a little loose in this regard, but I think to state it formally would add to the confusion rather than subtract from it. -- Fropuff (talk) 05:53, 23 February 2008 (UTC)
thread_title:  Definition of morphisms 

message: When forming a comma category over a 2-category and relaxing commutativity of the squares resp. triangles to the existence of a 2-cell, should we then speak of a lax comma category (resp. oplax comma category, if the direction of the 2-cell is against to that of the 1-cells)? --Tillmo (talk) 15:07, 16 June 2009 (UTC)
thread_title:  Lax comma category 

message: The page used to state, in the section "Category of objects under A" and several other places, that if T is a constant functor, then "fixing A makes α irrelevant", meaning that the comma category (T \downarrow S), where T is a constant functor with value A, is the same as 
(A\downarrow S). This is not so. For example, consider the case of
:\mathcal A \xrightarrow{\;\; T\;\;} \textbf{1}\xleftarrow{\;\; S\;\;} \mathcal B
where \textbf{1} is the terminal category (say with object *), and T and S are the unique (and constant) functors. It is immediately obvious from the definition that in this case the comma category (T \downarrow S) is isomorphic to \mathcal{A}\times\mathcal{B}. This is not at all the same as (* \downarrow *)\cong\textbf{1}.
I fixed this in today's edit. Did I miss something obvious? Selinger (talk) 06:39, 6 December 2010 (UTC)
thread_title:  Category of objects under A 

message: We might include a reference to a physical book, if these are considered more stable than on-line references. In the case of The Joy of Cats, the physical book (in the copy I have) refers the reader to http://katmat.math.uni-bremen.de/acc where the "newest edition of the file of the present book" can be downloaded. Which is the better template to use for cite, "Cite book" or "Cite web"? Model Math (talk) 17:45, 8 April 2017 (UTC)
thread_title:  Reference 

message: I'm wondering why the Comma Category turns into a Discrete Category when using A=B=1. Won't this be 'only' a Discrete Category of two selected objects (the ones selected by S and T)?
The text suggests that it is a general Discrete Category with all objects of C.
Unless we could consider a class of functor pairs (S,T) that selects all objects of C. (Varying  S and fix T would also do that job if I understand Category Theory right.) Ron van den Burg (talk) 09:19, 26 December 2023 (UTC)
replies: text: Never mind. The objects if the Comma Category are the morphisms from A to B (as stated in the artikel). And there can be many of them. All the morphisms of the Comma Category consist of (id_*, id_*) applied to all the Comma Category Objects, which are their identity morphisms. Ron van den Burg (talk) 11:23, 27 December 2023 (UTC)
thread_title:  Discrete Category