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Last June, Eduardo wrote at twisted arrow category:
you could view then morphisms from f to g as factorizations of g through f; this is in fact a good way of getting the arrows directions above right.
Eduardo, if you are reading this, or anyone else: can you explain further how this is supposed to help get the arrows’ directions right? Why should an arrow from f to g be a factorization of g through f rather than a factorization of f through g?
Is there a sense other than convention in which this direction of the arrows (as opposed to the opposite one, defining the category Tw(A)op) is the “correct” one? Is this the universally used convention?
I always think that the terminology ‘factorisation of f through g’ has a sense that you start with f and end up with g. (That suggests a direction for the arrow from f to g.) This is a bit like a subdivision; you subdivide f into three bits the middle one of which is g. The other way is like a composition. I think that Baues and Wirshing adopted the wrong convention for their terminology!
By the way Leech used this construction in his cohomology of monoids and Wells then worked with it for cohomology of categories before Baues and Wirsching came along. (I have added in the references into Baues-Wirsching cohomology. (Wells’ paper is very good. A pity it was not published.)
For me, it does help in the form of Fact(C)=Tw(C)op; I should have explained this bit.
Concerning directions, the main reason I see for Tw(C) as defined is that it is what you get from (*/homC); for factorizations, it looks fairly obvious to me that the more rational choice is Fact(C)=Tw(C)op
Eduardo, are you just saying that “morphisms from f to g are factorizations of g through f” helps you remember that the two arrows between f and g go in different directions, not which particular directions they go in?
was in a hurry yesterday, I will try to expand a bit on this:
First, there’s (at least for me) an obvious notion of Fact(C), where we want to have
Now, to match language usage (“f factorizes through g”), the direction of the morphisms should be
(a,b):f→g if f=bga
Let’s call this category Fact(C). Sadly, according to Tim #2 it looks like the term “category of factorizations” has been used to refer to Fact(C)op; anyway, I’ll stick for Fact(C) as defined for what follows.
Now, twisted arrow categories. For me, the definition is just tw(C)=*/homC; but to get an explicit description, it is easier for me to just remember tw(C)=Fact(C)op. This is essentially the content of “morphisms from f to g are factorizations of g through f”.
Lastly, about whether we should have tw(C)=Fact(C)op or tw(C)=Fact(C). I think that we should have tw(C)=Fact(C)op, because
Okay, thanks! I find your reason #4 the most compelling one.
Mike’s post about his new papers took me to twisted arrow category and then to Baues-Wirsching cohomology, neither of which I’d noticed before. Both seem strangely unlinked to the rest of nLab.
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