Talk:Glossary of category theory
I started this article, as there are many terms in category theory and the glossary article can come handy like many others. I know there are a good deal of overlaps right now but I think we can keep each main article (e.g., category (mathematics) focusing on more theorems and basic notions, and less on definitions and terminology. It is generally a bad idea to bombard readers with unfamiliar terms. -- Taku 07:05, August 6, 2005 (UTC)
a/the - language question
I'm not a native speaker, but:
- CAT is the quasicategory of all categories
sounds imho better than.
- CAT is a quasicategory of all categories
--Kompik 15:13, 20 February 2006 (UTC)
The book Abstract and Concrete Categories uses construct in the same meaning as concrete category is used in the glossary. (Construct is a concrete category over Set - Definition 5.1) --Kompik 15:13, 20 February 2006 (UTC)
All sections apart from the first are alphabetically sorted. I cannot see the reason why the items in the first section are ordered in this way. --Kompik 15:16, 20 February 2006 (UTC)
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A quasicategory is not a category
The article says "A category A is said to be: ... quasicategory provided that objects in A may not form a class and morphisms between objects A and B may not form a set". If the objects do not form a class and Mor(A,B) does not form a set, the thing is not a category. The definition of "quasicategory" should be moved out to its own paragraph. —Preceding unsigned comment added by 220.127.116.11 (talk) 18:10, 5 March 2008 (UTC)
- Hi. This is confusing because "class" means something different in different formalisms, and so "category" means something different in different formalisms. Herrlich and Strecker assume the collection of all classes form a "conglomerate". This is a reasonable foundation, but I don't think it is standard. I think the following is fair: What AHS call "category", others would call "large category"; what they call "quasi-category" corresponds roughly to what others would call "super-large category", or perhaps "category in the third Grothendieck universe". (I wonder if the "quasi" terminology may be too specific to warrant listing here at all.) Sam (talk) 15:45, 28 September 2008 (UTC)
- The discussion of "size" currently resides at Category of sets. Perhaps the AHS definition of "quasicategory" should be moved there… Sam (talk) 15:50, 28 September 2008 (UTC)
Would this article be more useful if it were in alphabetical order, as are Glossary of arithmetic and Diophantine geometry, Glossary of classical algebraic geometry, Glossary of differential geometry and topology, Glossary of Riemannian and metric geometry, Glossary of scheme theory, Glossary of topology and so forth? The reader who doesn't know exactly what a concept is has to scan the article: the editor who can't fit a topic in doesn't know where to put in. Deltahedron (talk) 18:18, 5 June 2014 (UTC)
- I just want to say this is a very good point. (I'm actually responsible for the current structure, but, well, I would say we know better now; from our experience the alphabetical order works better.) -- Taku (talk) 01:06, 16 June 2015 (UTC)
@TakuyaMurata: I removed the passage in question temporarily, since I believe that there is nothing to be shown and it is superfluous. If you disagree, please argue before reinstating the passage; we are mathematicians, so when in doubt, we can spell out the complete argument. Mine is that whether or not a category is preadditive or additive is a property by definition (since either a category satisfies the definition, or it doesn't, but certainly one of the two, and thus, preadditiveness or additiveness are or are not properties of the category). If I made a mistake, please tell me where I'm wrong (although I think I'm right). --Mathmensch (talk) 19:16, 29 January 2017 (UTC)
- @Mathmensch: Did you notice my response at my talkpage? The confusion stems from the fact that "additive" is an adjective even though, mathematically, it shouldn't be. What you can do is that you can consider a preaddirive structure on a category; it is not unique so it does not make sense to ask whether a category is preadditve or not. On the other hand, one can show that if there is a pre-additive structure and there are finite coproducts, then the pre-additive structure is unique; i.e., one can show additivity is a property of a category, which is to say you can ask a category is additive or not. The ref I gave should help you see the subtle difference (and has to be mentioned since it is tricky.) To repeat for emphasis, the first thing to notice is that it doesn't make sense to ask a category is pre-additive or not; the terminology here is a bit unfortunate and that's precise why we have this note. I added a clarification to emphasize this point. -- Taku (talk) 23:03, 29 January 2017 (UTC)