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This seminar by thermoplyae took place on 31st August 2008 20:00 UTC, in #mathematics channel of irc.freenode.net.

The timestamps displayed in the IRC text are UTC+1.
Any explanation or errors regarding this seminar should be recorded in the discussion page.

Topic Edit

Categories and Topology. Part of the series outlined at Towards Spectral Sequences

Seminar Edit

21:00:47 thermoplyae: alright, you set the channel topic, i grab a coke, and i guess we'll get
this started
21:01:23 ChanServ changed the topic of #mathematics to: SEMINAR IN PROGRESS.  If you want to
ask a question, say ! and wait to be called
21:01:49 _llll_: all systems go, or something
21:01:49 thermoplyae: okay then
21:02:27 thermoplyae: so this is the first in a series of seminars where the goal is to make it
to these rather complicated-to-the-point-of-mystical objects called
"spectral sequences"
21:03:50 thermoplyae: spectral sequences first showed up and found fantastic application in
algebraic topology by a couple of guys, Leray and Serre, where Serre
kicked Leray's into working over other things called "fibrations" and
allowed him to calculate the "cohomology ring" of a nice "space"
21:04:03 thermoplyae: the trouble, obviously, is that i can just tell you all what a spectral
sequence is without defining all those quoted words first
21:04:17 thermoplyae: and the majority of you are not algebraic topologists, so this
calculation would be 100% meaningless
21:04:52 thermoplyae: so that's what the series of seminars is for: introduce algebraic
topology, stretch our legs a bit, then look at the Serre and the
Atiyah-Hirzebruch spectral sequences to make a few cohomology calculations
21:05:17 thermoplyae: and then, if the audience is willing, we'll look at other examples of
spectral sequences so you can see the application outside of algebraic
topology proper
21:06:07 thermoplyae: and so i'm going to assume that you all are familiar with algebra (you
know what a group and a ring are) and that you are, to some extent,
familiar with topology (you know what a space is, how a continuous map is
defined, maybe a separation axiom)
21:06:53 thermoplyae: but, based on _llll_'s seminar, not too many of you are familiar with
category theory, so we ought to fill in a few gaps there and then outline
the first few categories we'll be working inside
21:06:55 thermoplyae: and that's the plan for today
21:07:45 thermoplyae: so, in case you've forgotten, a category is something like a directed
graph, where we have vertices called objects and edges called arrows
21:08:42 thermoplyae: they also come equipped with a mapping called "composition" that takes
two edges f: A -> B and g: B -> C and composes them into a single arrow
(gf): A -> C, and to each object A is attached a special arrow id_A that
satisfies (f id_A) = (id_B f) = f
21:09:58 thermoplyae: the name for a map of categories is a 'functor', and just like other
functions-with-extra-structure from algebra, a functor F is a function
that satisfies F(id_A) = id_(F A) and F(f g) = (Ff) (Fg)
21:10:34 thermoplyae: now, some material we didn't make it to in _llll_'s seminar, stripped down
somewhat:
21:11:20 thermoplyae: first, load this puppy up:

21:14:03 thermoplyae: so, given a functor
F: J -> C (here J and C are categories), the "limit of F" is an object
r that comes with maps r -> F_i for all objects in the image of F such
that the maps commute for arrows in the image of F.  in addition, for any
other object d that also has a family of arrows (think of it as a
candidate limit object), there has to be a unique map u: d -> r such that
all d's arrows factor through r's
21:14:11 thermoplyae: this is gibberish, so we'll do an example
21:14:40 thermoplyae: let's pick J to be a category consisting of two objects and whose only
two arrows are the identities on those objects
21:15:06 thermoplyae: a functor F from J into our working category C effectively picks out two
objects in C -- whatever is contained in its image
21:16:01 thermoplyae: so it's reasonable to think of F as a pair (a, b) of objects in C.  the
limit of F is some object r along with maps i'll call projection maps
r -> a and r -> b such that giving a map d -> r is exactly the same as
giving a pair of maps d -> a, d -> b
21:16:40 thermoplyae: this particular example of a limit is called a "product", and in the
familiar category of sets the product there is the cartesian product
21:17:21 thermoplyae: in groups it's the direct product, in abelian groups it's the direct
product, and in most familiar categories you can find familiar examples
with this fantastic property being able to relate maps into the product
with maps into its factors
21:17:46 thermoplyae: go for it
21:18:01 FunctorSalad: in groups the colimit is the free product
21:18:42 FunctorSalad: doesn't matter, just added it for completeness
21:19:55 thermoplyae: anyway, back on track, the important part is that -- again -- maps into
the product are the exact same as maps into each of its components
21:20:19 thermoplyae: we can do the same thing but instead of talking about maps into r we use
maps out of r
21:20:43 thermoplyae: the relevant diagram is:

21:21:47 thermoplyae: and for a given functor F: J -> C, blah blah blah, exact same thing with
all the arrows reversed, the object that satisfies the mapping-out
property (which i called s in the picture) is called the colimit of F
21:22:26 thermoplyae: again, picking J to be the category with exactly two objects and exactly
two arrows, functors F: J -> C still can be thought of as selecting two
objects in C
21:22:43 thermoplyae: and the colimit of that pair (a, b) is some object s where maps s -> d
correspond exactly with a pair of maps a -> d, b -> d
21:23:19 thermoplyae: so i'm told that the free product is the coproduct in the category of
groups, i know disjoint union is the coproduct in the category of sets,
and, again, plenty of other familiar examples exist
21:24:11 thermoplyae: another important example of a (co)limit is when the index category has
two objects a, b and maps id_a, id_b, f: a -> b, g: a -> b
21:25:29 thermoplyae: so functors J -> C look like pairs of objects with a pair of arrows
between them.  the (co)limit of such a functor is called a (co)equalizer.
in slightly less formal language, the equalizer of a pair of arrows is
the "largest" piece of the domain where the two arrows agree
21:25:55 thermoplyae: and the coequalizer is the least amount of work you have to do to the
codomain to make the two arrows agree in the end
21:26:18 thermoplyae: the plan, again, is to explore some topologically-relevant categories,
so no worries, we'll get to examples
21:28:01 thermoplyae: but, first, there's one last important kind functor i want to bring up:
representable functors
21:28:43 thermoplyae: which is something _llll_ covered, so this part should be intelligible.
given a category C, we can pick an object a and consider the set of
arrows a -> b for each object b in C
21:29:02 thermoplyae: i'll write C(a, b) for such a set
21:29:34 thermoplyae: given an arrow f: b -> c, there's a relation between C(a, b) and C(a, c)
-- specifically, there's a map C(a, b) -> C(a, c) given by
post-composing all the arrows in C(a, b) with f
21:29:51 thermoplyae: pictorially:

21:30:13 thermoplyae: this assignment is obviously functorial (seriously; if it isn't
immediately obvious, you should be able to think through it inside of a
few minutes)
21:30:44 thermoplyae: and so the assignment C(a, -) is a functor, called the functor
"represented by a"
21:31:29 thermoplyae: we can do something similar if a is sitting on the other side; there's a
relation between C(b, a) and C(c, a) induced by a map f: b -> c given by
precomposing all the arrows in C(c, a) with f to get an arrow in C(b, a)
21:31:39 thermoplyae: again, a picture may be helpful
21:32:19 thermoplyae: representable functors are sweet not just because they're easy to
describe, but because of the limit property i kept stating and restating
earlier
21:32:32 thermoplyae: let's say c is the product of a and b
21:32:51 thermoplyae: and, by definition, an arrow d -> c corresponds to a pair of arrows
d -> a, d -> b
21:33:24 thermoplyae: which describes an isomorphism of sets C(d, a * b) = C(d, a) * C(d, b),
where * is the product in the relevant category
21:34:11 thermoplyae: in fact, something more general is true (and is also pretty obvious, i
won't prove it): representable functors C(a, -) take limits in C to
limits in Set
21:34:31 thermoplyae: similarly, representable contravariant functors C(-, a) take colimits in
C to limits in Set
21:35:23 thermoplyae: not so bad, right?  let's talk about topology for a bit now
21:36:34 thermoplyae: and to justify all this category discussion, we ought to apply them a bit
in topology
21:37:06 thermoplyae: in case you've forgotten, a topological space (X, T) is a set X and a set
of subsets of X, where T satisfies
21:37:13 thermoplyae: 1) emptyset and X are in T
21:37:29 thermoplyae: 2) given any collection of elements of T, their union will also be in T
21:37:38 thermoplyae: 3) given any finite collection of elements of T, their intersection will
also be in T
21:38:03 thermoplyae: the sets in T are called the 'open' sets of X
21:38:05 thermoplyae: most of the time i'll say say 'X is a space' and omit T from the notation
21:38:53 thermoplyae: and a continuous map of spaces is a generalization of the continuous maps
you may have seen in basic analysis; a map f: X -> Y is continuous if
every open set in Y has a preimage that is open in X
21:39:50 thermoplyae: and so if we think of open sets as a description of how the space is
glued together, "open set" meaning something loosely like "region", this
is a fairly obvious restriction that says "our regions over here
have (Y) to be made up of regions from over there (X)"
21:40:02 Manyfold: !
21:40:07 thermoplyae: go ahead, manyfold

[at this point temporary network issues cause a short interval]

21:40:27 Manyfold: why finite collection with intersection?
21:42:14 Manyfold: and can you give an example were the collection is not a set
21:42:46 Manyfold: i ask cause my textbook on topology defines it as set of subsets

21:48:04 ~kees_: the second is easy to answer. The collection of open sets in question is
always a set.
21:48:40 ~kees_: the first question is a question of practicality.
21:49:11 ~kees_: if one tries to model the topology of R in terms of open sets then it's not
true that an arbitrary intersection of opens is open.
21:49:50 ~kees_: thus in the definition of a topology one does not require opens to be stable
under arbitrary intersections since this will exclude many examples.
21:49:54 Manyfold: ar as i understand collections are bigger than sets

21:52:17 ~_ar: as far as the topology here, you can safely take sets and collections to be
synonymous

3:42pm   thermoplyae: reasonable question, and i'm not sure i have a satisfying answer.
if we want to think of intervals (a, b) in R, those aren't closed under
arbitrary intersection, and maybe "we sure wish topology were
representative of things we already understand" is a good reason?
[3:43pm] thermoplyae: yeah, i don't know if a more solid answer exists; axioms are there
because they describe other things we're familiar with that we want to
generalize
[3:44pm] thermoplyae: an important thing you should immediately recognize is that the
composition of two continuous maps is continuous and that the identity
function is continuous
[3:45pm] thermoplyae: and so, tada, we have the first category we care about, where the objects
are topological spaces and the arrows are continuous functions.  we'll
call it Top, short for Topology
[3:46pm] thermoplyae: so, in an effort to feel out this category, we ought to look at what
some of those limits look like
[3:47pm] thermoplyae: the coproduct of two spaces A, B is given by the disjoint union of the
point sets underlying A and B and taking the open sets of the coproduct
to be sets whose intersection with A is open and whose intersection with
B is open
[3:47pm] thermoplyae: i don't have pretty pictures, but you ought to think of the two spaces
just being set down next to each other.  the coproduct of two circles
looks like... two circles
[3:48pm] thermoplyae: and the exact same definition works for infinite coproducts; open sets
are ones whose restrictions to the components are open
[3:49pm] thermoplyae: products are mildly more difficult to describe; the product of two spaces
A, B has point set given by cartesian product of the point
sets of A and B
[3:50pm] thermoplyae: if we pick an open set U n A and an open set V in B, we'll say U x V
to be open in A x B

21:56:00 thermoplyae: it's not enough that U x V be open in A x B; that collection isn't closed
under union
21:56:24 thermoplyae: and so we fix it by (surprise) closing it under union.  turns out that
what we get is the (binary) product topology
21:57:16 thermoplyae: for infinite products, we must be somewhat more careful; for the product
\prod_\alpha X_\alpha, we have finitely many U_\alpha be not all of
X_\alpha and the rest be their entire component spaces X_\alpha
(hopefully that's intelligible)
21:58:44 thermoplyae: we should also talk about quotient topologies, something those of you who
took undergrad topology likely have not seen, since that underlies much
of the rest of these constructions
21:59:26 thermoplyae: given a space X and a surjective map of sets f: X -> Y, we can construct
a "largest" topology on Y where V a subset of Y is open exactly when V's
preimage by f is open in X
22:00:14 thermoplyae: what does this mean, you ask?  well, let's take the interval [0, 1] with
the topology inherited from R (sets of the form [0, a), (a, 1],
and (a, b) are open for a < b in (0, 1))
22:00:47 thermoplyae: our equivalence relation will be the trivial equivalence relation
(a ~ a) augmented by the relation that 0 ~ 1
22:01:34 thermoplyae: and so everything sits in its own equivalence class except for one class,
which contains both 0 and 1
22:02:40 shminux: !
22:02:45 thermoplyae: yes
22:02:56 shminux: doesn't it turn [0,1] into a circle?
22:03:14 thermoplyae: it does, i was just writing that out as slowly as possible
22:04:04 thermoplyae: an explicit homeomorphism (fancy word for invertible continuous map) is
given by x |-> (cos (2 pi x), sin (2 pi x)), where the codomain is the
subspace of R^2 given by unit-length vectors

22:05:12 ~kees_: may I just remark that that is just one of many possible homeomorphisms
22:05:28 thermoplyae: oh, sure
22:05:38 ~kees_: the 'the' in the beginning of the sentence might imply there is just one.
22:05:52 thermoplyae: yes, sorry, good point
22:06:16 ~kees_: and you also have to demand that the inverse is cont as well, right?
22:06:32 thermoplyae: yeah, i mean "inverse in Top"

22:07:00 thermoplyae: so i did a sort of sleight of hand and identified surjective maps with
projections to equivalence classes while no one was looking
22:07:07 thermoplyae: this is also a profitable way to look at things
22:07:41 thermoplyae: given an equivalence relation on X, continuous maps out of X whose
values agree on the equivalence classes of ~ are exactly the same as
continuous maps out of X/~
22:07:53 thermoplyae: where X/~ has the quotient topology given by X -> X/~
22:09:04 thermoplyae: so back to categories, let's try to describe coequalizers
22:09:34 thermoplyae: we have two maps f, g: X -> Y, and we want a third map h: Y -> Z such
that hf = hg and it satisfies a universal property
22:10:18 thermoplyae: one obvious thing to do, now that we have this quotient topology, is to
quotient Y in such a way that f and g send points to the same equivalence
class
22:10:22 thermoplyae: i.e. f(x) ~ g(x)
22:10:54 thermoplyae: and the map Y -> Y/~ given by this equivalence relation is, in fact, the
pushout in Top (once again, proof is yours to handle)
22:11:44 thermoplyae: we can mix'n'match these constructions to form what are called pushouts,
another common form of limit
22:12:55 thermoplyae: we pick a pair of arrows f: X -> Y, g: X -> Z that share a common domain,
and we want to pick an object P with maps Y -> P, Z -> P such that the
square commutes and is universal (all other ways to fill out the lower
right corner factor through P and its arrows)
22:13:45 thermoplyae: the case we'll typically be interested in is the case that X is actually
a subspace of Y, and in this case P is written Y \cup_f Z
22:13:53 thermoplyae: another picture:

22:14:56 thermoplyae: and so what should we do?  well, we're going to have an arrow out of Y in
to P and out of Z into P, so we ought to start by taking the coproduct
of the two
22:15:11 thermoplyae: but that won't be enough; we'll have too many possible maps out of P
22:16:03 thermoplyae: in addition, we have to coequalize and fix it, effectively quotienting by
the relation f(x) ~ g(x), where f(x) lies in Y included into the
coproduct, g(x) lies in Z included into the coproduct
22:16:20 thermoplyae: so, what is this doing?  and why the notation Y \cup_f Z?
22:17:02 thermoplyae: let's pick some spaces to work with as examples; for instance, Y can be
a hemisphere, Z can be a plane, and X can be the boundary of Y, sort of
the equator
22:17:23 thermoplyae: the map X -> Z should just select some circle in the plane, not terribly
important
22:17:46 thermoplyae: we take the coproduct and then we quotient, stitching the points along
the equator of the hemisphere onto where we mapped them in the plane
with f
22:18:06 thermoplyae: and we ought to end up with a plane with a bubble sitting on top
22:18:22 thermoplyae: "Y \cup_f Z" should be read as "Y glued to Z along f"
22:19:21 thermoplyae: okay, so we have two more categories; we'll do Top^* because it's easy,
and Toph if there's time; i'll take a poll or something
22:19:23 thermoplyae: this is getting kind of long
22:20:07 thermoplyae: Top* is sort of a twist on Top; to each space X we associate a point
x_0 in X called the "basepoint" of X
22:20:25 thermoplyae: maps between spaces are still continuous maps, but we require that
f: (X, x_0) -> (Y, y_0) act as f(x_0) = y_0
22:20:39 thermoplyae: these spaces-with-basepoint are called "pointed spaces", and the maps
"pointed maps"
22:21:12 thermoplyae: the product looks exactly the same, where the basepoint of
(X, x_0) * (Y, y_0) is (x_0, y_0)
22:21:52 thermoplyae: the coproduct looks slightly different; we can no longer just take the
disjoint union of X and Y, since how would we know where our basepoint
went?
22:22:21 thermoplyae: indeed, we need maps out of X + Y to correspond to maps out of X and Y
separately, which maps the map out of X+Y has to determine the action
for both of their basepoints
22:22:37 thermoplyae: and so the right thing to do is (X disjoint union Y) / (x_0 ~ y_0)
22:23:57 thermoplyae: if we denote the set of all continuous maps X -> Y by Y^X, i'm going to
make the somewhat ridiculous-sound claim that, in Top, we can find a
topology on Y^Z such that Top(X x Y, Z) = Top(X, Z^Y) for "nice" spaces
X, Y, Z (something about being paracompact and hausdorff, don't think
about it too hard, we won't be focusing on pathologies)
22:24:37 thermoplyae: sadly, we can't transfer this fact directly to Top*
22:25:09 thermoplyae: in Top*, Z^Y has a basepoint given by the map that takes all of Y to the
basepoint of Z -- the constant map
22:25:49 thermoplyae: and so a pointed map X -> Z^Y will have to take the basepoint of x to
this constant map, in effect partially determining what ought to be a
map X x Y -> Z
22:26:10 thermoplyae: (specifically the part that shares a component with x_0)
22:26:36 thermoplyae: you can observe something similar going on with points in X x Y that share
a component with y_0 -- they also all have to get sent to the basepoint
in Z
22:27:13 thermoplyae: and so the right thing to do, obviously, is to take X x Y and quotient by
(X v Y), where X v Y is the coproduct of X and Y and the implicit
inclusion is along the subspace sharing a coordinate with the basepoint
of X x Y
22:27:45 thermoplyae: this operation is called the smash product, and it enjoys the property
that Top*(X smash Y, Z) = Top*(X, Z^Y), recovering the property that we
want
22:28:54 thermoplyae: alright, exhale.  are you guys up for homotopy and the homotopy category?
22:29:09 thermoplyae: if not, no worries, this is a reasonable place to stop
22:29:31 _llll_: i think people who are still following will still be following if you explain
that stuff
22:29:36 Cale: Adjunctions away! (sorry ;)
22:29:45 thermoplyae: excellent
22:30:12 thermoplyae: we won't do much because the moment we get done defining homotopy is the
moment we start doing algebraic topology, and that's too much for one day
22:31:18 thermoplyae: okay, so basically everything we'll ever do from this point on (including
the future seminars) will be in the context of Top*, with a few exceptions
22:31:57 thermoplyae: so, to start, when i write "I", i'll mean the unit interval [0, 1] where
the basepoint is 0, and when i write "X_+" i'll mean the space X + pt with
basepoint taken to be the point living in pt.
22:32:42 thermoplyae: we have two pointed maps f, g: X -> Y, and the most basic idea of
homotopy is that we can "slide" one over to the other
22:33:13 thermoplyae: what do we mean by that?  we mean there's a map H: X * I_+ -> Y such
that H(x, 0) = f(x) and H(x, 1) = g(x) for all x
22:33:42 thermoplyae: using the exponential law described above, we can think of this as a(n
unpointed) map I_+ -> Y^X, which is a path in the function space
22:35:15 thermoplyae: the relation "f is homotopic to g" is an equivalence relation on
Top*(X, Y), and composition of two representatives of two homotopy
classes Top*(X, Y) and Top*(Y, Z) all give elements of the same homotopy
class Top*(X, Z)
22:35:37 thermoplyae: and so we can consider "pointed spaces with maps distinguished up to
homotopy", a category denoted Toph
22:36:17 thermoplyae: an invertible morphism f: X -> Y in Toph is a homotopy class with
another homotopy class g: Y -> X such that fg is homotopic to id_Y and
gf is homotopic to id_X
22:37:04 thermoplyae: some more vocabulary includes maps that are homotopic to the constant
map X -> Y (i.e. lie in the same path component of Y^X), such maps are
said to be "null-homotopic"
22:38:04 thermoplyae: and, on a related note, if there's a homotopy equivalence (i.e. an
invertible homotopy class) between X and the one point space {*}
(i'm lazy about notation, i may at some point write * or pt), X is said
to be "contractible"
22:38:58 thermoplyae: and this leaves us on the brink of some actual work, so i think we'll
leave off
22:39:03 thermoplyae: any questions?  obvious mistakes? :)

22:39:52 ~kees_: I'd like to remark that you can do homotopy theory also for unpointed spaces
22:39:58 ~kees_: and for doubly pointed spaces
22:40:03 ~kees_: and for totally pointed spaces.
22:40:10 ~kees_: (and in many other categories).
22:40:17 thermoplyae: certainly
22:40:26 ~kees_: I'm not sure why you wanted to restrict to pointed spaces.

22:40:57 Cale: kees_: Possibly because he wants pi_1 to be a group valued functor later on?
22:41:07 Cale: (and not groupoid)
22:41:51 Cale: I really do like the fundamental groupoid though. The proof of the generalised Seifert-van Kampen theorem goes through more nicely, I think :)

22:41:02 thermoplyae: because it will make pi_n make sense
22:41:03 thermoplyae: yes

22:40:33 _llll_: you dont want to restrict
22:41:05 _llll_: a better approach would be to have a *set* of base-points
22:41:14 _llll_: ie E/Top not 1/Top=Top*

22:41:16 ~kees_: another remark is that Toph is actually not the thing you want to work with
(in some philosophical level) since you quotient out by homotipies
thereby loosing tons of information.

22:41:52 thermoplyae: well, from a strictly topological viewpoint, nothing in algebraic
topology is really what you "want" to be working with
22:41:54 ~kees_: Cale: could be, but the more modern approach of considering the fundamental
category seems better suited.

22:42:21 thermoplyae: but topology is so hard :( and we reduce to less complicated settings
where we can get things done
22:42:26 thermoplyae: "we" being "the people in #mathematics"

22:42:46 _llll_: i think the point is that the basepoint stuff doesnt make anything easier
22:42:49 ~kees_: thermoplyae: Joyal has a thing he calls HOT. It's a simplicial set (which is
a quasi cateogory) which is (so he claims) what you want to work with when
doing homotopy.
22:43:07 ~kees_: the point is that categories are too strict to handle such things. You wanna
weaken the notion of a cateogory.

22:43:37 Cale: _llll_: well, surely groups are a bit nicer things to have than groupoids.
22:43:47 _llll_: i dont think they are
22:43:48 ~kees_: not always Cale
22:43:52 ~kees_: in fact many times it's not.
22:44:04 _llll_: the coproduct of groups is one example
22:44:31 ~kees_: as categories Grpoid is much nicer and simpler than Grp (for example)
22:44:51 _llll_: there's some slogan that you want categories with nice properties, not
categories with objects with nice properties
22:45:15 FunctorSalad_: _llll_: yeah read that too recently. by Grothendieck IIRC?
22:45:21 Cale: _llll_: Coproducts for groups are okay, aren't they?
22:45:38 _llll_: depends what you mean by "ok"
22:45:47 ~kees_: right
22:46:02 _llll_: you can read "from groups to groupoids" by ronnie brown for more info

22:46:21 thermoplyae: anyway, cale guessed correctly that pointed spaces will be used in my
construction of the homotopy groups and reduced homology, and i wanted
to sweep basepoint notation under the rug
22:47:02 thermoplyae: it is certainly worth noting that homotopy makes sense elsewhere, and if
you're not going to do what i'm doing other things may be more
appropriate

23:32:08 _llll_: thermoplyae: what's next week's title?
23:33:42 thermoplyae: _llll_: homotopy and cell complexes?  i think that would work, and it sounds so unassuming
we may get as large a crowd as we did this week
23:34:03 _llll_: cool
23:35:01 |Steve|: Sounds good to me.

23:39:01 ChanServ changed the topic of #mathematics to: NEXT SEMINAR: Sunday 7 September 20:00 UTC:
Toward Spectral Sequences 2: Homotopy and Cell Complexes
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