David Yetter asked,
Is there an existing name in the literature for a category in which every cospan admits a completion to a commutative square? (Just that, no uniqueness, no universal properties required, just every cospan sits inside at least one commutative square). If so, what have such things been called? If not, does anyone have a poetic idea for a good name for such categories?
Eduardo Pareja-Tobes replied with quite a long discussion of filtered categories, but I am not sure whether he made the following point clear. Any group (the two-element one will do) considered as a one-object category has the property that David mentioned. However, it is not "filtered" in the sense of Chapter IX, Section 1 of Mac Lane's "Categories for the Working Mathematician". You have to look carefully at the printed diagram to see this. Whilst any (co)span in a group may be completed to a commutative square, a parallel pair does not have a common composite in this sense: -------> X Y -------> Z -------> David's definitive language suggests that he knows this and does not require this property of his categories, but other people who want to generalise ideas from posets to categories might fall into this trap. Maybe "amalgamation property" would be suitable for David, although usage of that term in the literature might require the maps to be monos. Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]