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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeMar 20th 2012

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 $\mathrm{Tw}\left(A{\right)}^{\mathrm{op}}$) is the “correct” one? Is this the universally used convention?

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeMar 20th 2012
• (edited Mar 20th 2012)

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.)

• CommentRowNumber3.
• CommentAuthoreparejatobes
• CommentTimeMar 20th 2012

For me, it does help in the form of $\mathrm{Fact}\left(C\right)=\mathrm{Tw}\left(C{\right)}^{\mathrm{op}}$; I should have explained this bit.

Concerning directions, the main reason I see for $\mathrm{Tw}\left(C\right)$ as defined is that it is what you get from $\left(*/{\mathrm{hom}}_{C}\right)$; for factorizations, it looks fairly obvious to me that the more rational choice is $\mathrm{Fact}\left(C\right)=\mathrm{Tw}\left(C{\right)}^{\mathrm{op}}$

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeMar 20th 2012

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?

• CommentRowNumber5.
• CommentAuthoreparejatobes
• CommentTimeMar 21st 2012

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 $\mathrm{Fact}\left(C\right)$, where we want to have

• objects arrows in $C$
• morphisms factorizations of one arrow through the other.

Now, to match language usage (“$f$ factorizes through $g$”), the direction of the morphisms should be

$\left(a,b\right):f\to g$ if $f=bga$

Let’s call this category $\mathrm{Fact}\left(C\right)$. Sadly, according to Tim #2 it looks like the term “category of factorizations” has been used to refer to $\mathrm{Fact}\left(C{\right)}^{\mathrm{op}}$; anyway, I’ll stick for $\mathrm{Fact}\left(C\right)$ as defined for what follows.

Now, twisted arrow categories. For me, the definition is just $\mathrm{tw}\left(C\right)=*/{\mathrm{hom}}_{C}$; but to get an explicit description, it is easier for me to just remember $\mathrm{tw}\left(C\right)=\mathrm{Fact}\left(C{\right)}^{\mathrm{op}}$. This is essentially the content of “morphisms from f to g are factorizations of g through f”.

Lastly, about whether we should have $\mathrm{tw}\left(C\right)=\mathrm{Fact}\left(C{\right)}^{\mathrm{op}}$ or $\mathrm{tw}\left(C\right)=\mathrm{Fact}\left(C\right)$. I think that we should have $\mathrm{tw}\left(C\right)=\mathrm{Fact}\left(C{\right)}^{\mathrm{op}}$, because

1. $\mathrm{tw}\left(C\right)=*/{\mathrm{hom}}_{C}$
2. In the $\mathrm{Set}$ case, ends reduce to limits with shape $\mathrm{tw}\left(C\right)$, while _co_ends reduce to colimits over $\mathrm{tw}\left(C{\right)}^{\mathrm{op}}$
3. Making $\mathrm{tw}\left(C\right)$ a functor in $\mathrm{Cat}$, $f$ is a unique lifting of factorizations functor iff $\mathrm{tw}\left(f\right)$ is a discrete fibration
4. it has been consistently defined as such in the past, in all of the sources I’ve seen
• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeMar 21st 2012

Okay, thanks! I find your reason #4 the most compelling one.