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Hi everyone. In a regular category, we have a good notion of the ’image’ of a morphism: it is given by the coequalizer of the cokernel pair of the morphism. In particular, if we apply this to a morphism in the category Set, which is regular, we obtain the usual notion of the image of a function.
I’m interested in finding a good notion of the image of a functor. The most straightforward approach, it seems to me, would be to weaken the process of constructing images in a regular category, to obtain generalized ’images’ of 1-cells in a ’regular 2-category’. But now we should be taking the pseudo-coequalizer of the pseudo-cokernel pair. (See John Power’s article “2-Categories” for a good introduction to these sort of limits.) The 2-category Cat has all pseudo-limits. Is it ’regular’ in the correct sense? If so, what notion of the image of a functor results from this approach? Does this give us a factorization system for Cat?
Here’s a fun example, which any good notion of the ’image’ of a functor should be able to handle. Let $F:\mathbb{Z}_2 + \mathbb{Z}_2 \to S_3$ be the functor sending each copy of the group $\mathbb{Z}_2$ to the subgroup $\{ \mathrm{id}, (12) \}$ in $S_3$, and let $G:\mathbb{Z}_2 + \mathbb{Z}_2 \to S_3$ be the functor sending the first $\mathbb{Z}_2$ to the subgroup $\{ \mathrm{id}, (12) \}$ and the second $\mathbb{Z}_2$ to the subgroup $\{ \mathrm{id}, (13) \}$. I’m treating the groups $\mathbb{Z}_2$ and $S_3$ as one-object categories. The functors $F$ and $G$ are isomorphic, and so should certainly have equivalent images, and I want that image to be equivalent to $\mathbb{Z}_2$. But note that only $F$ actually factorizes through $\mathbb{Z}_2$; $G$ doesn’t. A naive notion of the ’image’ of $G$ might be the smallest subgroup of $S_3$ containing the subset $\{ \mathrm{id}, (12), (13) \}$, which would be all of $S_3$ - this is not what I want!
Mike Shulman has written about these sorts of things in his nLab pages. In particular, there is interesting material on regular 2-categories and factorization systems. He talks about the (eso+full,faithful) factorization system for regular 2-categories, which seems close to the sort of thing I’m looking for. It seems plausible to me that you could define the ’image’ of a morphism relative to a given factorization system, as the ’smallest’ object through which the morphism factors. However, I can’t see how this could deal in a non-evil way with the difference between the functors $F$ and $G$ described above.
I would be interested to hear any comments that anybody has.
Jamie.
There is the notion of essential image of a functor.
There is also the notion of homotopy image which for functors makes sense in the context of the folk model structure.
I’m a little confused. I would tend to say that the ’image’ of a morphism relative to some factorization system is simply the factorization of that morphism – isn’t that what a factorization system is for, to factor a morphism into its image?
Regarding your example, I don’t see how F and G are isomorphic. Two group homomorphisms are isomorphic as functors when they are conjugate, but it seems that anything you use to conjugate one copy of (12) into (13) would also conjugate the other.
And actually, I don’t even understand the definition of G. The elements (12) and (13) don’t commute in $S_3$, so how can they both be in the image of an abelian group?
Could it be that Jamie’s ’$+$’ is the free product (the coproduct in $Grp$)? That would take care of the definition of $G$.
Urs: thanks for pointing that out, I didn’t know the term ’essential image’. I think this would fail my evilness test, though, as it would assign inequivalent images to isomorphic 1-cells in Cat, such as the functors F and G described above. I’m not sure about the homotopy image - it’s too late at night for me to start thinking about infinity-things! Can it be applied to plain old Cat, somehow?
Mike: thanks for your comments. The problem I see with getting images from a factorization system is that a particular morphism can have multiple factorizations, and in general the factorizing objects will not be isomorphic (or equivalent, if we’re in a 2-category.) Of course, you get perfectly good images from the friendly (epi,mono) system in Set - but I don’t see how you can get well-defined images from the (eso+full,faithful) system in Cat, for example.
Perhaps I’m making a silly mistake, but I think that $F$ and $G$ are indeed congugate. The elements (12) and (13) aren’t in the image of an abelian group, they’re in the image of an abelian groupoid - the domain of both $F$ and $G$ is the groupoid $\mathbb{Z}_2 + \mathbb{Z}_2$. Let me know if the definitions of F and G still aren’t clear. There is a natural transformation $\alpha:F \Rightarrow G$, which takes the value $\mathrm{id} \in S_3$ on the first copy of $\mathbb{Z}_2$ and the value $(321) \in S_3$ on the second copy of $\mathbb{Z}_2$.
Todd: Yes, I’m working in the 2-category Cat, not the 2-category of groups. For the case of the example, of course, I could just as well be working in the 2-category of groupoids.
groupoid $\mathbb{Z}_2 + \mathbb{Z}_2$
spelling it out: the groupoid with two objects….
Ah, I see, your + means the coproduct of groupoids. Another good reason to use a notation like “B” for the functor that regards a group as a one-object groupoid; if you’d written $B \mathbb{Z}_2 + B \mathbb{Z}_2$ then I wouldn’t have gotten it confused with $B(\mathbb{Z}_2 + \mathbb{Z}_2)$.
It is always true, in any orthogonal factorization system (E,M), that the factorization of any morphism into an E-map followed by an M-map is unique up to unique isomorphism. This follows from the orthogonality of E and M (and actually, the converse is also true – uniqueness of factorizations implies orthogonality). The same is true for 2-categorical factorization systems: factorizations are unique up to equivalence. In particular, this is the case for the (eso+full, faithful) factorization system, which more recently I’ve been calling the (2-epic, 2-monic) factorization system.
The (eso+full, faithful) factorization of any functor $C\to D$ can be obtained by starting with C and identifying all pairs of parallel morphisms which become equal in D. In the particular case of your functors $F,G\colon B\mathbb{Z}_2 + B\mathbb{Z}_2 \to B S_3$, this means that nothing gets identified, so in both cases the image is just $B\mathbb{Z}_2 + B\mathbb{Z}_2$ again. This is what they have to be, because the functors F and G are both already faithful – just like the image of an injective function is (isomorphic to) itself.
Finally, as remarked on the page essential image, it’s not clear that that notion is really well-defined for arbitrary functors. I think this functor is one for which it doesn’t make sense; in particular, it is not pseudomonic.
Can it be applied to plain old Cat, somehow?
As I said, one could try to use the notion (of homotopy image) with the canonical model structure on $Cat$.
I haven’t thought about this, I just thought I’d mention what the $n$Lab has on notions of higher images.
a remark/question:
the definition of image of a morphism $f : c \to d$ as
$im f := \lim_{\leftarrow} (d \stackrel{\to}{\to} d \coprod_c d)$has an evident generalization to higher categorical contexts: take the limit to be the relevant notion of higher limit ($(2,1)-limit$, say) and then set
$im f := \lim_{\leftarrow} (d \stackrel{\to}{\to} d \coprod_c d \stackrel{\to}{\stackrel{\to}{\to}} d \coprod_c d \coprod_c d \cdots) \,.$What happens if we apply this inside Grpd?
Mike: I’m confused, because $F$ can also be written as the composite of $[\mathrm{id},\mathrm{id}] : B\mathbb{Z}_2 + B\mathbb{Z}_2 \to B\mathbb{Z}_2$, which is eso+full, and the embedding $B\mathbb{Z}_2 \to B S_3$, which is faithful. Does this not give a nonequivalent factorization?
Urs: I think applying that definition of image inside Grpd is a good idea! I think that’s essentially what I was suggesting in the original post, although I don’t understand the details of what you’ve written. What’s a ’(2,1)-limit’? How come the expression for the image of the morphism involves an infinite chain?
I tried yesterday to work out the pseudo-coequalizer of the pseudo-kernel-pair of $F$, and I got $B\mathbb{Z}_2$, but I’m not very good with higher limits, and it’s perhaps suspicious that I got the answer I wanted! So, maybe someone else should have a go.
Mike said:
The (eso+full, faithful) factorization of any functor $C \to D$ can be obtained by starting with $C$ and identifying all pairs of parallel morphisms which become equal in $D$.
Another problem I have with this is that if we restrict to discrete categories, it doesn’t reproduce the familiar (epic,monic) factorization we would expect by viewing the discrete categories as sets, and the functors between them as functions.
However, it seems to me that the embedding Set$\to$Cat, obtained by adding trivial 2-cells to Set and regarding its objects as discrete categories, carries (co)limits into pseudo-(co)limits. So if we factorize $2 \to 1$ into $2 \to 1 \to 1$ in Set, we have to do the same thing in Cat, or we can’t claim to have a consistent approach.
I think applying that definition of image inside Grpd is a good idea! I think that’s essentially what I was suggesting in the original post,
Okay, I think that must be the way to find out what the right answer is, even if that right answer then may turn out to have a more explicit description in terms of something else, like factorization systems. These should be models for something more intrinsic.
although I don’t understand the details of what you’ve written. What’s a ’(2,1)-limit’?
I just mean the notion of weak limit in $Grpd$. That happens to be a (2,1)-category, so I wrote $(2,1)$-limit to indicate that. So effectively that should be what is described at 2-limit, only that that page does not really state a definition. I am thinking of it as an $(\infty,1)$-limit aka homotopy limit in groupoids. Whatever terminology you like best.
How come the expression for the image of the morphism involves an infinite chain?
That’s because I was indicating how it should look in $\infty Grpd$. In $Grpd$ you can termminate this chain and consider
$im f := \lim_{\leftarrow} (d \stackrel{\to}{\to} d \coprod_c d \stackrel{\to}{\stackrel{\to}{\to}} d \coprod_c d \coprod_c d) \,.$Urs, this is still a bit mysterious to me. Why would we not expect to be able to calculate the image of a 1-cell in a 2-category as the equalizer of the cokernel pair $\lim_{\leftarrow} (d \stackrel{\to}{\to} d \coprod_c d)$? Why do we need the limit of a more complicated diagram?
I haven’t really thought about this for images, but that’s what turns out to be the right answer in many other cases:
the sheaf condition is an equalizer, the $\infty$-sheaf condition is the $\infty$-limit over the simplicial diagram of which the equalizer diagram was only the first step.
a regular epimorphism is the coequalizer, of its parallel pair of kernel pair maps, a regular $\infty$-epimorphism is the $\infty$-colimit over its full Cech nerve diagram, of which the two parallel arrows are onle the first stage.
And so on.
Jamie,
here is a better reply as to why we need that full cosimplicial diagram:
with that, the image inclusion will precisely be a regular monomorphism in the higher category sense (discussed there).
Now, here is a good way to see why we have these cosimplicial diagrams in the definition of regular monomorphisms: they are dual to regular epimorphism, where we have simplicial diagrams instead, namely Cech nerves. These are the generalization of what in the 1-categorical case is a kernel pair . A kernel pair is an equivalence relation. In higher category theory, the equalities in this equivalence relation are resolved to isomorphisms in a groupoid and further to equivalences in higher groupoids. The Cech nerve of a morphism is the $\infty$-groupoid of its kernel pairs. This appears in the def of regular epimorphisms. Here for our regular monimorphism inclusions of images, we see the dual co-Cech nerves.
I added my propossed def of $(n,1)$-image here.
Thanks Urs, that’s very interesting. Unfortunately, it’s all rather opaque to me - I won’t be able to appreciate it until I understand why this is the ’right’ definition of regular epimorphism. Also, I would hope that all of this applies not only to images in Grpd, but images in Cat as well, so I don’t want to have to rely on being in an (n,1)-setting.
Also, I think it’s important to remember that we’re talking about images of functors - something which should be very basic and fundamental! If this simplicial diagram is the correct thing to use here, we should be able to understand and motivate it at this level.
I need help: I’m trying work out a general recipe for calculating co-iso-inserters in Cat, but I’m failing miserably. We need this to try out some of these suggestions for the image of a functor. Iso-inserters aren’t so bad, but co-iso-inserters are thorny, in just the same way as coequalizers are trickier than equalizers in Set. Can anyone give me a hand?
We talked about related things around here, where Mike gave an explicit description of the objects of the iso-inserter.
Also, I think it’s important to remember that we’re talking about images of functors - something which should be very basic and fundamental! If this simplicial diagram is the correct thing to use here, we should be able to understand and motivate it at this level.
Sure, go ahead. I know now what the right answer is, so when you can come up with a guess, I can check. ;-)
Probably I said tis the wrong way, let me try again: I don’t know the explicit answer, but I do know an algorithm that pretty certainly gives the right answer. One would just need to sit down and unwind it.
For instance: the definition I give evidently satisfies the two desiderata that you added to image: it sends equivalent morphisms to equivalent images and on discrete groupoids it reduces to the ordinary notion of image of sets.
Also, I would hope that all of this applies not only to images in $Grpd$, but images in $Cat$ as well, so I don’t want to have to rely on being in an (n,1)-setting.
This is a curious attitude that I encounter a lot in discussions: faced with a difficult problem and a solution to a nontrivial but tractable special case, that solution is ignored on the basis that it is not yet the fully general answer. I think that’s not a good attitude. Instead, understanding the solution to the special case is likely to be a big clue in finding the solution to the general case.
Just a remark. But ’ll shut up now.
Yes, I agree - but they would also be satisfied by the naive image, which just takes the weak coequalizer of the weak kernel pair. We need to know the difference between this naive approach I’m suggesting, and the more sophisticated one you’re talking about. I find these weak colimits too difficult, at the moment, otherwise I would just jump in and calculate to get a feel for these things.
I find these weak colimits too difficult
So if I had time to think about this, I would do the following: I would restriczt attention to functors between groupoids first. Then I would compute the weak pushouts as homotopy pushouts using the standard model structures as described in some detail in the list of concrete examples at homotopy limit.
So this always works in the same kind of way: to compute a pushout diagram of
$A \leftarrow C \to B$you find equivalences (of groupoids in our case) $A \stackrel{\simeq}{\to} \hat A$ etc such that we get a diagram
$\array{ A &\leftarrow& C &\to& B \\ \downarrow^\simeq && \downarrow^\simeq && \downarrow^\simeq \\ \hat A &\leftarrow& \hat C &\to& \hat B }$and such that in the bottom row all objects are cofibrant and at least one of the two morphisms is a cofibration.
So that means you find, if necessary, equivalent but somewhat thickened versions of your three groupoids.
Then you compute the ordinary pushot (in the 1-category of groupoids). The result of that is then the (uniquely defined up to equivalence) weak pushout.
The same for pullbacks and limits, with all morphisms reversed.
So if I were you I’d open a text on canonical model structure on $Cat$ and do such example computations.
I think once you get going with a few examples, this becomes an easy and somewhat fun exercise. As I said, maybe warm up with looking at some of the worked examples spelled out at homotopy limit/homotopy pullback.
Urs, I’m not doing that at all - it’s great to have a reduction to a particularly tractable special case! In fact, what I said had the opposite meaning to that which you read into it - I didn’t say that I wanted to ignore your solution, but rather that I had hopes it would extend to a more general setting.
Just to be clear: my point is that if the image in the form $\lim_{\leftarrow} (d \stackrel{\to}{\to} d \coprod_c d \stackrel{\to}{\stackrel{\to}{\to}} d \coprod_c d \coprod_c d) \,.$ works well in Grpd, it’s not crazy to think it might also work in Cat.
Thanks very much for your suggestions about homotopy colimits - that is immensely enlightening, and I shall have a go!
Jim Dolan pointed out to me some years ago that in a $2$-category, you don’t want to have a $2$-step factorisation anyway, but a $3$-step one. (Later maybe you can compose two of those steps if you really want to.) In particular, in $Cat$, you want the factorisation
source → weak coimage → full image → target
which is explained at the end of this old thing (3-page pdf).
In particular, if the category happens to be a set, then the weak coimage and the full image are both the same as the ordinary image. (By the way, I think that I made up the term “weak coimage” myself, and I don’t particularly care about it.)
I don’t have time to read through all this discussion now! But I will say quickly that the fold map $\nabla\colon B Z_2 + B Z_2 \to B Z_2$ is not full, because if x and y are the two objects of $B Z_2 + B Z_2$, then there are morphisms from $\nabla(x)$ to $\nabla(y)$ in $B Z_2$ (such as the identity) which aren’t in the image of any morphism from x to y in $B Z_2 + B Z_2$. So the factorization you suggested in #11 is not actually an (eso+full, faithful) factorization.
Another problem I have with this is that if we restrict to discrete categories, it doesn’t reproduce the familiar (epic,monic) factorization we would expect by viewing the discrete categories as sets, and the functors between them as functions.
Yes. This is one of the reasons I think it’s questionable to call a faithful functor a “subcategory” and the (eso+full, faithful) factorization its “image.” I think it’s more reasonable to apply those words (unprefixed) to the (eso, full+faithful) factorization system, and put a 2- on them for the (eso+full, faithful) case. So the 1-image of a functor is a full subcategory (= 1-subcategory) of its target, which extends the notion of ordinary image for sets as discrete categories, and the 2-image is something bigger (a faithful functor = 2-subcategory), which doesn’t.
Of course, for group homomorphisms regarded via B as functors, it is the 2-image which extends the ordinary notion of image for group homomorphisms. This is another reason why it’s good to use a visible notation for B – it’s a “delooping” operation which shifts things up a level, so it’s not surprising that it turns 1-images into 2-images.
Jamie, here’s a more concrete explanation of why you need the whole (co)simplicial diagram. The equalizer of a pair of parallel functions $f,g\colon A\rightrightarrows B$ is the subset of those elements of A which satisfy a property: namely that $f(a)=g(a)$. However, the “pseudo-equalizer” of a pair of parallel functors $f,g\colon A\rightrightarrows B$ is a category whose objects are objects of A equipped with extra structure, namely an isomorphism $f(a) \cong g(a)$. But in general, it’s a bad idea to ask for a category equipped with isomorphisms that don’t satisfy any coherence equations! E.g. in a monoidal category, we require the pentagon and unit laws, similarly the hexagons for a braided/symmetric monoidal category, and so on. Including the next stage of the cosimplicial object is what gives us a coherence equation for the isomorphisms in this case; in the sheaf world, this is the “cocycle equation” for descent in a stack. In dimensions >2, then the next stage would just give us an isomorphism/equivalence sitting in the cocycle triangle, which would then want to satisfy its own equation up to higher equivalence, and so on, necessitating use of the whole cosimplicial object.
One other thing is that in general, taking equalizers of cokernels (pseudo or not) doesn’t usually give you very good image factorizations; coequalizers of kernels is usually what you want to do. In the higher case, that means the “codescent object” of the “Cech nerve” which is a simplicial diagram. In Cat or Gpd, this prescription gives you the (eso, full+faithful) factorization system, and more generally this works in any regular 2-category (michaelshulman).
Toby: Those notes are great! Wow, this mysterious Jim Dolan does come up with lots of funky stuff.
Mike: Thanks for pointing out my dumb fullness mistake. I like your idea about (eso,full+faithful) giving us the ’1-image’ and (eso+full,faithful) giving us the ’2-image’, and I agree with the usefullness of the $B$s notation. It’s good to see where the (eso,full+faithful) factorization comes from, abstractly, in a regular 2-category.
So, do you know what sort of image you get when you construct the pseudo-coequalizer of the pseudo-kernel of a functor? Or what sort of image you get when you use the once-extended simplicial diagram instead? Now that Urs has told me about homotopy limits I can have a go at working this out myself, but perhaps you know already, or can make a good guess.
As I said in #28 (though perhaps not very clearly), if you take the colimit (= codescent object) of the simplicial kernel (= Cech nerve), you get the (eso, full+faithful) factorization. I don’t know what you get if you truncate the simplicial kernel after one stage (“pseudo-coequalizer”), but I’d expect it not to be very interesting.
Wow, this mysterious Jim Dolan does come up with lots of funky stuff.
Hear, hear! Mysterious indeed, and he is one of the most remarkable mathematicians you could ever meet.
OK. I misinterpreted your comments in #28 as I was under the impression that your article on regular 2-categories concerned strict limits, rather than pseudo ones.
So, it seems that the result of all this is that we still don’t know of a good factorization system on Cat or Grpd which factorizes $F:B \mathbb{Z}_2 + B \mathbb{Z}_2 \to B S_3$ as defined above into
$B \mathbb{Z}_2 + B \mathbb{Z}_2 \to B \mathbb{Z}_2 \to B S_3,$which is really what I’m hoping for. Since neither of the functors in the factorization are full, that rules out the typical factorizations we’ve been talking about.
One other thing is that in general, taking equalizers of cokernels (pseudo or not) doesn’t usually give you very good image factorizations; coequalizers of kernels is usually what you want to do.
This is coimage factorization, though.
So I suppose then Jamie needs to think about if he strictly wants a notion of image, or if the factorizaiton property is more important.
Why is it that coimages give good factorization systems and images not?
here’s a more concrete explanation
I am tempted to say that this is effectively what I did say in 16…
I had more trouble giving a good interpretation of the cosimplicial diagram, so I explained it via dualiztation.
I made a silly mistake in #25, but that’s OK, because it makes things better.
So the 3-way factorisation is (eso+full,eso+faithful,full+faithful). If the source is a set, then the eso+full part is an equivalence (not the eso+faithful part as I stupidly said), so we should call the (eso,full+faithful) partial factorisation the $1$-image and the (eso+full,faithful) partial factorisation the $2$-image (which Mike already said in #26).
source → $2$-image → $1$-image → target
which, incidentally, means that a target is a $0$-image and a source is an $\infty$-image.
But what do we call the morphisms? Specifically, what do we call the eso+faithful functor from the $2$-image to the $1$-image? The situation in a regular $1$-category offers no advice.
I realized that my comments #28 and 30 might have been misleading; in a (2,1)-category it makes no difference, but in a (regular) 2-category in order to get the correct factorization you have to be sure to use the right notion of kernel and codescent object, which involve noninvertible 2-cells. The kernel involves a comma object rather than a (pseudo) pullback, and likewise the codescent object.
My personal impression is that “coimages” and “images” got named the wrong way ’round. But I don’t know a really good explanation for why the one is more useful than the other. It’s essentially the same problem as why plain monomorphisms are useful, but plain epimorphisms are usually the wrong thing to think about unless they happen to coincide with some stronger type of epimorphism.
I am tempted to say that this is effectively what I did say in 16
I wasn’t referring to the dualization; I think the same arguments apply to either the situation or its dual. My point was to further emphasize and expand on your sentence “the equalities in this equivalence relation are resolved to isomorphisms in a groupoid and further to equivalences in higher groupoids,” which I think is the real explanation and might have been missed. (In fact, I missed it the first time I read your comment! (-: )
Jamie, can you say anything about why you want a kind of factorization which would produce $B Z_2 + B Z_2 \to B Z_2 \to B S_3$? Are there particular important properties of that factorization that we might be able to generalize?
Or, could you give us some other factorizations which fit the same schema? For instance, suppose G and H are two non-conjugate subgroups of a group K; what would you like the factorization of $B G + B H \to B K$ to be?
Urs: Regarding #33, I’m interested in the entire factorization of a functor $A \to B$ into an isomorphic functor $A \to \mathrm{Im} \to B$. And regarding #34, having multiple levels of redundancy is always good when trying to learn about something!
Toby: I like this systematic approach - although I’m disappointed it doesn’t seem to give what I’m looking for :(.
More later on Mike’s questions, after I’ve thought a bit more.
My original motivation for asking about this comes from representation theory. Given a groupoid homomorphism $f: C \to D$, we can obtain the restriction functor $f^*: \mathrm{Rep}(D) \to \mathrm{Rep}(C)$. By the Doplicher-Roberts theorem, this gives an equivalent perspective on the theory of groupoids, which is pretty profound and important. My social life is so relentless I have to head off in a few minutes to a party, so I don’t have time to explain more about this right now - if someone wants me to, I can explain later on a bit more about how this works.
Anyway, let’s say you’re happy about the monoidal functor $f^*: \mathrm{Rep}(D) \to \mathrm{Rep}(C)$ being equivalent to the groupoid homomorphism $f: C \to D$. Then instead of trying to factorize $f$, we can try to factorize $f^*$. This is a problem with a totally different feel! Here’s one way to do it: we can take all of the objects in $\mathrm{Rep}(C)$ which are in the image of $f^*$, and form the sub–symmetric monoidal category of $\mathrm{Rep}(C)$ which we can build from these objects using all the operations we’re allowed to do in a group representation category, like take duals of object, apply symmetry natural isomorphisms, and take equalizers. Call this new category $\mathrm{Im}$. Then we have a factorization of $f^*$ of type $\mathrm{Rep}(D) \to \mathrm{Im} \to \mathrm{Rep}(C)$ in a natural fashion, and going back to the groupoid-centred worldview instead of the representation theory-centred worldview, we obtain a factorization of our original groupoid homomorphism $f$.
So, I can tell you immediately what the factorization system is, from this perspective: it’s $f \simeq f_2 \circ f_1$, where $f_1^*$ is essentially injective, and $f_2^*$ is ’dense’, in the sense that all the objects in its target category are colimits of diagrams in the image of $f_2^*$. The challenge is to understand what these properties are directly in terms of $f_1$ and $f_2$ as algebraic operations, rather than in terms of what they do to representations. The property on $f_1^*$ probably boils down to $f_1$ being ’jointly surjective’, or something, meaning that all the morphisms in its target groupoid are in its image, up to isomorphism. I have no idea what the density condition should boil down to.
Interesting. What makes you believe that the image category Im is also of the form Rep(E) for some groupoid E?
Your description of the construction you have in mind reminds me of the (hyperconnected, localic) factorization of a geometric morphism. I think that that is constructed by taking the closure of the image of the inverse-image functor $f^*$ under subobjects and quotients. Moreover, for geometric morphisms between presheaf topoi induced by functors between the domain categories, the (hyperconnected, localic) factorization does indeed correspond exactly to the (eso+full, faithful) factorization.
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