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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2010

    added to image

    • an expanded Idea-section (check!)

    • the evident proposal for the definition of (,1)(\infty,1)-categoical images here (which implies in particular a notion of images of of nn-functors between nn-groupoids).

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJul 22nd 2010

    Image of a category appears as a part of a factorization system on Cat. Can this help to get an alternative check on your approach ?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2010

    Yeah, somebody should work out what the 2-limit that I am proposing actually boils down to.

    But: it’s clearly the right definition. ;-)

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJul 22nd 2010
    • (edited Jul 22nd 2010)

    I meant the factorization systems of that kind have refinement into (n+1)-step factorization systems which are categorical analogues of the Postnikov tower, as alluded in the lectures of Baez and Shulman. I find that fascinating…I wish somebody finishes this idea from their lectures with hard theorems.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2010
    • (edited Jul 22nd 2010)

    categorical analogues of the Postnikov tower, as alluded in the lectures of Baez and Shulman. I find that fascinating…I wish somebody finishes this idea from their lectures with hard theorems.

    We have hard theorem about the notion of Postnikov tower in an (infinity,1)-category.

    The definitions of that, at least, have immediate analogs for (,n)(\infty,n)-categories.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeJul 22nd 2010
    • (edited Jul 22nd 2010)

    I know there is much about general Postnikov towers in (infinity,1)-setting. But I was talking about having specifically the statement that there is a (n+2)-step factorization system on the category of weak (n,n)-categories, and that this factorization system is related to some case of Postnikov construction in (infinity,1)-setting. My understanding is that the existence of that (n+2)-step factorization system is known for strict n-categories. In Baez-Shulman paper this is alluded in the treatment of (n,k)(n,k)-surjectivity. Thus I am looking for a full solution to a very concrete problem alluded in Baez-Shulman exposition. In fact, in written literature, there is no full treatment of the factorization system for strict n-categories, but it is probably just the matter of writing or at worst an undergraduate diploma project.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2010
    • (edited Jul 22nd 2010)

    My understanding is that the existence of that (n+2)-step factorization system is known for strict n-categories. In Baez-Shulman paper this is alluded in the treatment of (n,k)-surjectivity.

    My understanding is the the notion of k-surjective functor in their article is the globular reformulation of what simplicially is a morphism that has the right lifting property against the kkth generating cofibration

    Δ[k]Δ[k]. \partial \Delta[k] \to \Delta[k] \,.

    Namely a functor between strict globular \infty-categories is kk-surjective precisely it if it has the right lifting property against the inclusions of the boundary of the kk-globe into the kk-globe

    G kG k \partial G_k \to G_k

    (See here.)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2010
    • (edited Jul 22nd 2010)

    And essentially kk-surjective is something like surjective on the kkth categorical homotopy groups, where the kkth-categorical homotopy group is images of G kG_k or Δ[k]\Delta[k] with constant boundary modulo translation along equivalences.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeJul 22nd 2010

    Right. Now the question is if one can make a factorization system out of those as sketched in Baez-Shulman.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 23rd 2010

    I have an idea how to do this. Take for a test case 1-categories. For a category CC map XOb(C)X \to Ob(C) let C[X]C[X] be the category given by the pullback codisc(X)× codisc(Obj(C))Ccodisc(X)\times_{codisc(Obj(C))} C (In other words, objects XX and arrows X 2× Obj(C) 2Mor(C)X^2\times_{Obj(C)^2} Mor(C)). Then for a functor f:DCf:D \to C there is a factorisation

    DC[Obj(D)]C[im(f 0)]C D \to C[Obj(D)] \to C[im(f_0)] \to C

    where f 0f_0 is the object-component of ff.

    A similar game can be played with 2-categories, where one can not only define C[X]C[X] for a set XX, but also a reflexive graph X =(X 1X 0)X_\bullet = (X_1 \rightrightarrows X_0) with composition (something like a (pointed magma)-oid :) and a map of such things X C 1X_\bullet \to C_{\le 1} for the underlying 1-category C 1 C_{\le 1} of a 2-category (this works for bicategories too, but let’s stick to strict things). In particular, for the underlying category D 1D_{\le 1} of a second 2-category DD equipped with a 2-functor F:DCF:D \to C. In general this is a bicategory, but if X X_\bullet is a category, then C[X ]C[X_\bullet] is a 2-category.

    We can then define

    DC[D 1]C[D 0]C[im(F 0)]CD \to C[D_{\le 1}] \to C[D_{\le 0}] \to C[im(F_0)] \to C

    Each of the things C[?]C[?] expresses that 2Cat2Cat is fibred (opfibred?) over various other categories and they can all be defined in terms of pullbacks with variously codiscrete 2-categories(1). This should make it manifest what sort of things are forgotten (stuff, structure etc) at each step. The universal property of the pullbacks helps ensure the uniqueness up to isomorphism of the factorisation.

    The general pattern should be clear. There is a functor codisc n+1:nCat(n+1)Catcodisc_{n+1}: nCat \to (n+1)Cat adding to each category a unique arrow between any two parallel n-arrows. For an nn-functor DCD \to C define C[D m]=codisc m+1(D m)× codisc(C m)CC[D_{\le m}] = codisc_{m+1}(D_{\le m}) \times_{codisc(C_{\le m})} C where the (m+1)(m+1)-codiscrete (m+1)(m+1)-categories are considered nn-categories with only identity arrows between dimensions m+1m+1 and nn. Here m=0,,n1m=0,\ldots,n-1. The successive truncations together with the universal property of the pullback should furnish the functors in the factorisation.


    (1) There is a functor codisc 2:rGrp compBicatcodisc_2:rGrp_{comp} \to Bicat from reflexive graphs with composition to BicatBicat by adding to a reflexive graph a unique 2-arrow between any two parallel 1-arrows. Coherence happens automatically. The restriction of codisc 2codisc_2 to CatrGrp compCat \hookrightarrow rGrp_{comp} lands in 2CatBicat2Cat \hookrightarrow Bicat.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 23rd 2010

    The simplicial version of this should be by using coskeleta. Actually I think the above can be expressed more succinctly by by considering the codiscrete categories of the truncations as steps in the Postnikov tower for DD (not really a Postnikov tower except for groupoids, but you know what I mean. It is easy to check these form a tower, where each map in it is taking equivalence classes of the top-dimensional morphisms): and then follow the formal construction of the Moore-Postnikov tower in TopTop from the Postnikov tower of one of the spaces involved (can’t remember off the top of my head how this is done, but it’s easy).

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2010
    • (edited Jul 23rd 2010)

    The page image says that its various definitions are equivalent when they jointly apply. But I think this is false, since the first two (“in terms of subobjects” and “as a left adjoint functor”) deal only with monomorphisms, while the third (“as an equalizer”) always produces a regular monic.

    I think the problem is that the notion of “image” really depends on first fixing a notion of subobject, which is different in different contexts. I prefer the first two definitions at image, which can easily be relativized to any notion of “subobject”, and are closely related to factorization systems. On the other hand, the third definition I view as simply a construction of images in the particular case when “subobject” means “regular mono.” I know that some people use “coimage” and “image” to mean specifically quotients of kernels and coquotients of cokernels, but I think I would prefer to view both of those as constructions of particular kinds of images.

    Back on the topic of everyone else’s discussion… I don’t quite understand what the question is. Is it whether there is (what I would now prefer to call) a (strong k-epic, k-monic) factorization system on strict n-categories? Or on weak n-categories (what sort)? Or whether such a factorization can be constructed using limits and colimits in some higher-categorical way?

    (By the way, I’m currently thinking of using “k-full” for what was called “essentially k-surjective” in Baez-Shulman. The problem with “k-surjective” is that a functor of this sort isn’t literally surjective on k-morphisms, even up to equivalence – only “locally” so in a sense which exactly generalizes the ordinary notion of “full functor.” In particular, the functor out of the empty category is k-full for all k>0. Also, “essentially k-surjective” could be argued to violate the rule that a “1-thing” should be the same as a “thing,” since we often say “essentially surjective” for “essentially surjective on objects”, i.e. 0-full. Thoughts?)

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 23rd 2010

    I think Zoran was wondering if the details of the ’layer cake philososphy’ of multi-stage factorisation had been written down for n-functors, in particular, for strict functors and n-cats as a warm-up. One is lead to wonder: what is the definition of an nn-step factorisation system on a category? One could require uniqueness up to various compatible isomorphisms…

    I claim my prescription in #10 would work for weak nn-categories, I think the only thing to show is that there is a strict pullback of nn-cats, and that the operation ’identify parallel top-dimensional arrows in an nn-category’ gives an (n1)(n-1)-category and that this is functorial.

    I like the idea of k-full. For 1-categories ’2-full’ should mean ’faithful’, for the same reason that a locally full 2-functor between 1-categories is just a faithful 1-functor. But one could modify and say things like (at the cost of syllables!) essentially locally kk-surjective or some permutation. Or we could throw an ’essentially’ in there, with essentially kk-full. Or we could go the other way and take kk-full as the weak notion, and use strictly kk-full for the actually surjective on parallel kk-morphisms.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2010

    I definitely want “k-full” to mean the up-to-equivalence notion by default. Yes, of course 2-full for 1-categories means faithful, and so on.

    Is the factorization DC[ob(D)]C[im(f 0)]CD \to C[ob(D)] \to C[im(f_0)] \to C supposed to be the 3-step factorization of a functor? It looks to me like the two categories in the middle are equivalent and both give the (eso, ff) factorization.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 23rd 2010
    • (edited Jul 23rd 2010)

    The page image says that its various definitions are equivalent when they jointly apply. But I think this is false,

    I was wondering about this a few times, but never did anything.

    So what do we do? You are the one with most thoughts on this, so I think it would be good if you edited the entry according to your comments in #12.

    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeJul 23rd 2010
    • (edited Jul 23rd 2010)

    I am not sure about your present notation, but I have checked in all details at some point the 3 step factorization on strict categories and 4 step factorization on strict 2-categories, after reading your paper with John. On the other hand, I do not recall that i cleared out the full axiomatic definition of what n-step factorization systems are, though some wanted properties are immediate. David correctly understood my question (on the other hand he organizes the construction proposal in 10 a bit different than what I remember so I need to rethink carefully; I hope to return to that next week – I have due the Arnold paper for local radio next week).

    • CommentRowNumber17.
    • CommentAuthorTobyBartels
    • CommentTimeJul 23rd 2010

    (By the way, […] Thoughts?)

    I agree.

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 23rd 2010

    supposed to be the 3-step factorization of a functor?

    yes…. I thought the middle two categories weren’t equivalent, but now I see they are. hmm. I thought I’d correctly interpreted Toby’s notes to something I understand, but I’ll have another go at it.

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2010

    The best definition I can think of for a 3-step factorization system is simply a pair of ordinary factorization systems (E 1,M 1)(E_1,M_1) and (E 2,M 2)(E_2,M_2) such that E 1E 2E_1 \subseteq E_2 (and hence M 2M 1M_2 \subseteq M_1). I think that condition implies that if you factor a morphism into (E 1,M 1)(E_1,M_1) and then factor the M 1M_1 part into (E 2,M 2)(E_2,M_2), you get the same thing as if you first factored into (E 2,M 2)(E_2,M_2) and then factor the E 2E_2 part into (E 1,M 1)(E_1,M_1), giving you the 3-step factorization. In the case of categories, E 1=E_1= eso+full, M 1M_1 = faithful, E 2E_2 = eso, and M 2M_2 = full+faithful. Probably for a (k+1)-step factorization system you similarly want k nested ordinary factorization systems.

    • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeJul 23rd 2010

    Signs of run-away negative thinking: within a minute of reading Mike’s last comment, I decided that nn-step factorisation systems make sense down through n=1n = -1 in a way that fits in perfectly.

    • CommentRowNumber21.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 24th 2010

    I figured out where I went wrong: instead to factoring the rightmost functor in

    DC[D 0]C D \to C[D_0] \to C

    I should factor the leftmost functor. Given a functor f:ABf:A \to B which is the identity on objects, we can factor it as Aim MorfBA \to im_{Mor} f \to B where Obj(im Morf)=Obj(A)=Obj(B)Obj(im_{Mor} f) = Obj(A)=Obj(B) and Mor(im Morf)Mor(im_{Mor} f) is the image in Mor(B)Mor(B) of the arrow component of ff. This fits the pattern I proposed better, in that the factorisation for 2-functors would then use the (epi,mono) factorisation in Set on 2-arrows, then the (eso,ff) factorisation in Cat on the level of 2- and 1-arrows, and then finally the (eso,local equivalence) factorisation in 2Cat.

    In Toby’s polynomial notes the 3-step factorisation is not the same, but is equivalent to mine via the equivalence that Mike noted above, so at the very most we need uniqueness up to equivalence.

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeJul 24th 2010

    I reorganized image along the lines I had in mind in comment 12. What should we do about coimage? I would kind of like to redirect it to image, so people get to the more general discussion; thoughts?

    I would also like to propose that instead of “3-step” or “3-stage” factorization systems we use the word ternary, and similarly ordinary factorization systems are binary and general ones are k-ary. The prefixes “3-step” and “3-stage” confuse me as to whether they mean the factorization has three intermediate morphisms or three intermediate objects, but the words “ternary” and “binary” remind me that we are writing f as a ternary, resp. binary, composite of things, so that the number is the number of morphisms. Thoughts?

    I can guess that a unary factorization system would be no structure at all (factor every morphism into a unary composite, namely itself). And that a nullary factorization system would factor every morphism into a composite of no things, i.e. an identity morphism – so that a category admits a nullary factorization system iff it is discrete. But I don’t know what a minus-unary factorization system would be; does it make the category empty? Contractible?

    • CommentRowNumber23.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 24th 2010

    we use the word ternary, and similarly ordinary factorization systems are binary and general ones are k-ary

    I agree.

    As far as the definition of a k-ary factorisation system, is it possible to say the naive thing, and say that it is an ’section of k-ary composition’? Then a ’functorial k-ary factorisation’ is something like functorial binary factorisation being a section of (in the notation of Emily Riehl’s ’concise definition of a model cat’) the functor d 1:C 3C 3d_1:C^\mathbf{3} \to C^\mathbf{3} – or is this too strong?

    We could additionally require that for every partition k=l+mk = l + m the composites of the first ll and the last mm arrows gives us a binary factorisation. Or we could be unbiased and say that for every partition k=m 1++m jk=m_1 + \ldots + m_j we get a j-ary factorisation system by forming the m im_i-ary composites… It would be nice to get this as a theorem from a basic description, though.

    • CommentRowNumber24.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 24th 2010
    • (edited Jul 24th 2010)

    (accidental copy of previous post)

    • CommentRowNumber25.
    • CommentAuthorTobyBartels
    • CommentTimeJul 24th 2010
    • (edited Jul 24th 2010)

    @ Mike #22

    I’d be inclined to say that ‘nn-step’ implies nn morphisms while ‘nn-stage’ implies nn objects. And actually, I’d prefer to count the objects! But I will adopt your terminology.

    For n>1n \gt 1, I claim that an nn-ary factorisation system consists of n +n^+ (that is n+1n + 1) factorisation systems (E i,M i)(E_i,M_i) (for 0in0 \leq i \leq n) such that

    • M iM i +M_i \subseteq M_{i^+} for 0i<n0 \leq i \lt n (equivalently, E iE i +E_i \supseteq E_{i^+} for 0i<n0 \leq i \lt n),
    • M 0M_0 consists of only isomorphisms/equivalences (equivalently, E 0E_0 consists of all morphisms), and
    • M nM_n consists of all morphisms (equivalently, E nE_n consists of only isomorphisms/equivalences).

    (Or course, an nn-ary factorisation system is determined by the n1n - 1 factorisations systems (E i,M i)(E_i,M_i) for 0<i<n0 \lt i \lt n, but the the other two exist.) Do you agree?

    Given an nn-ary factorisation system, the (co)image of (E i,M i)(E_i,M_i) is the ii-(co)image of the entire nn-ary factorisation system. (This agrees with the terminology in CatCat for n=3n = 3, or more generally with the terminology in (n2)Cat(n - 2) Cat or even (,n2)Cat(\infty,n - 2) Cat.)

    Then extending this definition to lower values of nn, every category (or \infty-category) has a unique 11-ary factorisation system, where (E 0,M 0)(E_0,M_0) is (iso,all) and (E 1,M 1)(E_1,M_1) is (all,iso), as you suggested.

    A category has a 00-ary factorisation system if and only if it is a groupoid, in which case (E 0,M 0)(E_0,M_0) is both (iso,all) and (all,iso) at once. In other words, rather than requiring every morphism to be a 00-ary composite on the nose, we require every morphism to be a 00-ary composite up to isomorphism. I think that this is right, since a factorisation system (of any arity) should be given by specifying full and replete subcategories of the arrow category (or equivalently, collections of isomorphism classes of the arrow category), and every isomorphism is isomorphic to an identity.

    I was wrong to say that n=1n = -1 fit; the definition above does not actually make sense in that case, since the conditions required of the 00 factorisation systems don’t parse. However, it seemed obvious to me that the empty category has a (1)(-1)-ary factorisation systems, since we require an impossible condition of every morphism. But maybe that is a bad intuition.

    Another thing that we should define is an \infty-ary factorisation system; Cat\infty Cat has one of these. This consists of \infty factorisation systems satisfying the first two conditions of nn-ary factorisation systems, but there is no room for the last condition. Note that Cat op\infty Cat^op has an op\infty^op-ary factorisation system.

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeJul 24th 2010

    Do you agree?

    Perhaps; that’s certainly a natural definition. I haven’t verified that it actually gives you a uniquely defined n-ary factorization of every morphism, as I have in the case n=3, but hopefully it’ll work out. That may be equivalent to David’s suggestion of giving a section of n-ary composition together with assuming that whenever n=k+n=k+\ell we get a binary factorization system by composing the first kk and the last \ell morphisms in the n-ary factorization.

    I think I agree with your version of 0-ary factorizations as well. Not sure about (-1)-ary ones, though.

    I also think this would be a good topic for a discussion at the Cafe. Perhaps I’ll start a post there.

    • CommentRowNumber27.
    • CommentAuthorTobyBartels
    • CommentTimeJul 24th 2010

    Perhaps I’ll start a post there.

    If you do, please announce it here, since I don’t look at the Café much these days.

    • CommentRowNumber28.
    • CommentAuthorTobyBartels
    • CommentTimeJul 24th 2010
    • (edited Jul 24th 2010)

    Another thing that we should define is an \infty-ary factorisation system […]

    OK, I think that I’ve got that now. This also clarifies what an (1)(-1)-ary factorisation system (called 00-stage below) should be.

    Fix any ordinal number (or opposite thereof, or any poset, really) α\alpha. Then an α\alpha-stage factorisation system (in an ambient \infty-category CC) consists of an α\alpha-indexed family of factorisation systems in CC such that:

    • M iM jM_i \subseteq M_j whenever iji \leq j (equivalently, E iE jE_i \supseteq E_j whenever iji \leq j),
    • each morphism f:XYf\colon X \to Y is both the inverse limit limiim if\underset{i \to \infty}\lim \im_i f in the slice category C/YC/Y and the direct limit colimicoim if\underset{i \to -\infty}\colim \coim_i f in the coslice category X/CX/C, and
    • for each f:XYf\colon X \to Y, id Y\id_Y is colimiim if\underset{i \to -\infty}\colim \im_i f and id X\id_X is limicoim if\underset{i \to \infty}\lim \coim_i f.

    Results to check:

    1. For every α\alpha-indexed family of factorisation systems that satisfies the first of the three axioms above, there is a unique (1+α+1)(1 + \alpha + 1)-stage factorisation system with the original factorisation systems in its middle, and every (1+α+1)(1 + \alpha + 1)-stage factorisation system is of this form.
    2. In particular, a 33-stage factorisation system is equivalent to a factorisation system, and there is always a unique 22-stage factorisation system.
    3. A 11-stage factorisation system exists iff CC is a groupoid, in which case it is unique.
    4. An nn-ary factorisation system from #25 is the same as an (n+1)(n + 1)-stage factorisation system.
    5. A 00-stage factorisation system exists iff CC is discrete, in which case it is unique.
    6. An α op\alpha^op-stage factorisation system in CC is the same as an α\alpha-stage factorisation system in C opC^op.
    7. The factorisation systems of Cat\infty Cat based on levels of fullness form an ω\omega-stage factorisation system in Cat\infty Cat.

    The last result sets the numbering and also shows that we really do want examples where α\alpha is unbounded (important since the first result shows us how the definition can be made much simpler when α\alpha is bounded).

    • CommentRowNumber29.
    • CommentAuthorMike Shulman
    • CommentTimeJul 28th 2010
    • CommentRowNumber30.
    • CommentAuthorTobyBartels
    • CommentTimeJul 28th 2010

    Thanks!

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2017

    The pages image and (epi,mono) factorization system and also abelian category knew nothing of each other. I have added some cross-links.

    • CommentRowNumber32.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 15th 2017
    • (edited Jun 15th 2017)

    image did not mention the concept of extremal epimorphism until about midway into the article. In contrast, extremal epimorphism says

    An image factorization of a morphism ff is, by definition, a factorization f=mef = m \circ e where mm is a monomorphism and ee is an extremal epimorphism.

    where the boldface has been introduced for this comment only.

    Readers might therefore expect extremal epimorphism to already appear within section “2. Definition” of image and be a bit puzzled that it does not.

    Therefore added a reference to [extremal epimorphism, mediated by some slight reformulation.

    Incidentally, removed the parenthetical “(then f=mef = m e)” which while maybe useful to remind readers of “Leibnizian” order of composition, seems too much here.

    Readers’ experience remains slightly more rough than necessary, on account of the proviso “if […] C$ has equalizers” in the section Relation to factorization systems).

    • CommentRowNumber33.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 15th 2017
    • (edited Jun 15th 2017)

    I think you have introduced confusion in the article image by not paying close contextual attention to the fact that it had been written for a general notion of (MM)-image where MM is a given class of monomorphisms. That is, extremal monomorphism makes reference just to the default meaning of “monomorphism”. I’m going to roll back to version #39 (edit: #40, I mean) and see if appropriate changes are nevertheless warranted.

    It also seems to me the proviso “has equalizers” is appropriate. I’m not sure why you think you speak for “readers” generally. If you have questions about the mathematics of an article, then please ask.

    • CommentRowNumber34.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 15th 2017
    • (edited Jun 15th 2017)

    I added a parenthetical “(MM)-” to make the context clearer.

    I looked at the parenthetical “(i.e., f=mef = m e)” and I personally find it unobtrusive. In addition, it may well be that the author of that parenthetical has experience of other readers who would actually find that helpful. Let’s leave it be.

    Edit: I also made a footnote at extremal epimorphism, clarifying which notion of image is meant.

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