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Stream: learning: questions

Topic: What is this factorisation?

Amar Hadzihasanovic (Jan 26 2024 at 13:47):

Suppose I have a category $C$ with

• an orthogonal factorisation system $(E, M)$, where
• every object $c$ is equipped with a colimit cone $\gamma_c$ that presents it;

for example, $C$ could be a presheaf topos with the epi-mono factorisation system and the canonical presentation of presheaves as colimits of representables.

Amar Hadzihasanovic (Jan 26 2024 at 13:49):

Given any morphism $f: c \to d$, it seems to me that, in this situation, I can obtain a factorisation of $f$ relative to the presentation of $c$, whose second leg is “locally $M$”, in the following way:

Amar Hadzihasanovic (Jan 26 2024 at 13:50):

1. Take the cone $f \circ \gamma_c$ over $d$, and factorise each component (at some index $j$) into $m_j \circ e_j$

Amar Hadzihasanovic (Jan 26 2024 at 13:50):

2. By functoriality of the factorisation, $j \mapsto \mathrm{cod}(e_j) = \mathrm{dom}(m_j)$ extends to a diagram, whose shape is the same as the shape of $\gamma_c$: let $f_*(c)$ be the colimit of this diagram, presented by the cone $\eta$.

Amar Hadzihasanovic (Jan 26 2024 at 13:51):

3. We get a morphism $\tilde{f}: c \to f_*(c)$ by universality of $\gamma_c$ considering the cone $j \mapsto \eta(j)\circ e_j$, and a morphism $f_l: f_*(c) \to d$ by universality of $\eta$ considering the cone $j \mapsto m_j$

Amar Hadzihasanovic (Jan 26 2024 at 13:52):

4. We have $f = f_l \circ \tilde{f}$.

Amar Hadzihasanovic (Jan 26 2024 at 13:52):

Question is: Have you seen this before? Does it have a name? Does it have nice properties e.g. in the presheaf topos case?

Amar Hadzihasanovic (Jan 26 2024 at 13:58):

One concrete example: take $C$ to be the presheaf category of reflexive graphs. Then I believe that, for any morphism $f$,

• $\tilde{f}$ will be a morphism that can identify adjacent vertices & also “contract edges”, but cannot identify non-adjacent vertices or non-contracted pairs of edges;
• $f_l$ will only identify non-adjacent vertices and other pairs of edges.

Amar Hadzihasanovic (Jan 26 2024 at 14:02):

To give my intuitive understanding of this: $f_*(c)$ is constructed by “gluing the images through $f$ of the representables in $c$, exactly in the way that they are glued together in $c$”

James Deikun (Jan 26 2024 at 15:22):

Hm, it seems like this wouldn't generally yield an EM factorization system since you could repeat the process on $f_l$ (there's no guarantee that the cone $\eta$ is the canonical one for $f_*(c)$)...

Amar Hadzihasanovic (Jan 26 2024 at 17:40):

Yes, I would not expect that to work in general.

Perhaps I should add that, in the particular example that I am interested into,

• the canonical cone is from a diagram of “atomic” objects, but also
• the image of an atomic object through any morphism is an atomic object,

so in fact $f_*(c)$ is, while not literally the canonical cone, in a certain sense 'equivalent' to it (sorry for the vagueness but I'm still figuring this stuff out); and in this case the factorisation does seem to be functorial.

Amar Hadzihasanovic (Jan 26 2024 at 17:43):

But meanwhile, I would be also just happy to have some understanding of how to characterise the “first leg” morphisms of this factorisation. In the example I care about the second leg is really characterised by being “locally $M$”, and that fits my intuition about the general construction, but I don't have a good understanding of what the first leg is. It seems like it should belong to $E$ in any realistic case, but beyond that I don't know.

Amar Hadzihasanovic (Jan 31 2024 at 09:46):

Just as an update.
I have focussed on the case that I was originally interested into, which is the category of posets and closed (i.e. lower set-preserving) order-preserving maps, where each poset $P$ is “canonically” presented as the colimit of the $P$-shaped diagram of embeddings of lower sets of its elements. The OFS is just epi-mono, which in this case is (surjective closed order-preserving map, closed embedding).

In that case, it turns out that this does produce another interesting orthogonal factorisation system:

• the right class is the local embeddings, that is, maps that restrict to closed embeddings on each lower set;
• the left class is maps that for now I've called upward-connected, which are maps that are surjective and satisfy the following property: if $f(x) \leq f(y)$, then there exists a zig-zag $x \leq x_1 \geq x_2 \leq \ldots \geq x_m \leq y$ such that $f(x) \leq f(x_i)$ for all $i$.

Amar Hadzihasanovic (Jan 31 2024 at 09:49):

The middle object arising from the factorisation of $f: P \to Q$ is the colimit of the $P$-shaped diagram of images through $f$ of the lower sets of elements of $P$.

Amar Hadzihasanovic (Jan 31 2024 at 09:50):

I'd be curious to know both if this OFS is known, and if it has any interesting version for categories instead of posets!

Amar Hadzihasanovic (Jan 31 2024 at 10:40):

The obvious generalisation of the right class would be functors $F: C \to D$ which induce equivalences $C/x \to D/Fx$ on each slice; the obvious generalisation of the middle-object is the Grothendieck construction of the functor $C \to \textbf{Cat}$ sending $x$ to $D/Fx$; not sure what the characterising property of the left class becomes, nor what the "correct" generalisation of closed maps is -- is it full & essentially surjective, or only essentially surjective on all slices?

Morgan Rogers (he/him) (Jan 31 2024 at 11:05):

Sounds related to the [[comprehensive factorization system]]?

Amar Hadzihasanovic (Jan 31 2024 at 11:40):

Yes, must be! Discrete fibrations between posets are precisely the local embeddings

Amar Hadzihasanovic (Jan 31 2024 at 11:41):

And I think I can roughly see how upward-connected is equivalent to "final" when a map is also closed...