Mention the ULF factorisation system.

]]>Added a reference to double categories.

]]>Re #28:

Even easier, use the alternative Markdown syntax: typeset as [link text][1] and then

1: https://doi.org/…

]]>Either escape the parenthesis as “%28” for “(” and “%29” for “)”

Thanks!

]]>Add cross reference to enhanced factorization system.

]]>The DOI has a parenthesis in it which is breaking the URL format. Is there a way to get around this?

Either escape the parenthesis as “`%28`

” for “(” and “`%29`

” for “)”

or (e.g. if it’s too much trouble remembering these numbers)

fall back to using the HTML tag “`<a href="url">text</a>`

” (which does not choke on parenthesis in URLs)

The DOI has a parenthesis in it which is breaking the URL format. Is there a way to get around this?

]]>Added an early reference.

]]>Cite Introduction to Homotopy Theory

]]>Cite introductory texts.

]]>Added a link to strict factorization system.

]]>Add reference to Mac Lane’s “bicategories”.

]]>Typo fix

]]>Made a link to a new page “comprehensive factorization system”.

]]>Added this observation

]]>Orthogonal factorization systems are equivalently described by the (appropriately defined) Eilenberg-Moore algebras with respect to the monad which belongs to the endofunctor $\mathcal{K} \mapsto \mathcal{K}^2$ of (the 2-category) Cat.

I could have done this myself, but I must confess I didnt manage to - and I am writing on a smartphone!

Best,

Beppe. ]]>

move the 2-categorical examples to a separate section, explicitly marked.

]]>Oh I see. Being bijective-on-objects doesn’t imply being essentially-bijective-on-objects. I got the wrong idea from the fact that being surjective-on-objects does imply being essentially-surjective-on-objects.

]]>Being able to lift up to isomorphism is *much* weaker. One will, I believe, need the axiom of choice to get the essentially surjective, fully faithful 2-factorisation. Just think about equivalences of categories.

The factorisation of $F: A \rightarrow B$ can be taken to $A \rightarrow C \rightarrow B$, where $C$ is defined by $Ob(C) = Ob(A)$ and $Hom_C(a,b) = Hom_B(F(a), F(b))$, where the two functors are the obvious ones.

One needs bijectivity on objects rather than just surjectivity to be able to construct a lift (given $F_1 \circ F_0 = F_3 \circ F_2$ where $F_1$ is fully faithful and $F_2$ is bijective on objects, one constructs the lift on objects as the inverse of $F_2$ on objects composed with $F_0$ on objects; on arrows one uses the inverses coming from the fully faithfulness of $F_1$). Uniqueness is obvious.

]]>Well at the moment it has both (eso, fully faithful) and (bo, fully faithful). They can’t both be correct, since the left class should be determined by the right. But the page doesn’t say that it’s only giving examples in 1-categories, so (because Cat is nicer as a 2-category than a 1-category) I think it would be better to just give (eso, fully faithful) and (eso+full, faithful), rather than whatever their strictified versions are.

I think I’m missing something though. Why is bijective-on-obejects the correct analogue of essentially-surjective-on-objects (rather than just surjective-on-objects)? How can every functor (in particular a functor not essentially-injective-on-objects) factor as (bo, fully faithful) when both bo and full imply essentially-injective-on-objects?

]]>Hi Beppe, welcome, and thanks for the question! Yes, I think you’re right. Instead of eso and eso+full, I think one needs to strictify a bit, to bijective on objects and bijective on objects + full respectively. Let’s fix this, and maybe give a few more details. I’ll not get to it just now; maybe someone else can give it a go?

]]>I was reading the page

https://ncatlab.org/nlab/show/orthogonal+factorization+system

and at the "Examples" section I read that among the classical examples of OFS in Cat, there are:

(i) Cat, (eso, fully faithful) factorization system

(ii) Cat, (eso+full, faithful) factorization system

mmm... aren't those examples of factorization system in a 2-cat?

Beppe. ]]>

Yes, I agree with your PS that the entry is correct; we have to allow the projections to be invertible rather than just identities in order to include all the isomorphisms in both classes. It’s certainly possible to have an OFS in which some of the factorizations are “extra-special” in a way not noticed by the factorization system.

]]>