This is the blog that documents my project of learning about
∞-categories. I've been looking at various references,
but currently I'm following a set of lecture notes that I found on
D.Cisinski's webpage. Basically, I'm rewriting his notes to suit
my taste, and in the process I also add all the exercices that he
left to the reader, as well as some further complements, and I correct
some mistakes. It's a big project : I've only started in the summer of
2022, and I started as an absolute beginner, so it's still very
tentative and constantly evolving.
Your comments are welcome and gratefully acknowledged!
Status Update :
November 15, 2022 : Cisinski has pointed
out to me a simple proof, attributed to Myles Tierney and found in
Joyal's CatLab
that shows that weak equivalences in any model category are stable
under retracts.
Once one has such result, one can quickly prove Whitehead's theorem,
following the argument in Hovey's book. I've included these proofs;
as a consequence, the complicated discussion of equivalences of
homotopies (that I borrowed from Quillen) is no longer needed, so
I've removed it... But I might reinstate it in the future, if it
becomes useful again : what else is it good for? I only used it to
prove Whitehead's theorem, but the new proof of the latter is much
simpler... Let me know, if you're an expert who happens to stumble
on my notes...
February 24, 2023 : today I'm finally done
with the part about the "classical" theory of Kan-Quillen, that endows
the category of simplicial sets with a model category structure whose
fibrant objects are the Kan complexes; all this is relevant to ∞-categories,
since Kan complexes are reinterpreted later as ∞-groupoids... Next up
is Joyal's model structure on simplicial sets, for which the fibrant
objects are exactly the ∞-categories. But for now I am taking
some rest, because I'm slowed down by teaching, and also because I
plan to flesh out the first preliminary chapter, with extra material
on calculus of fractions and the subobject classifiers for categories
of presheaves.
June 10, 2023 : There is an interesting
byproduct of homotopy techniques, concerning the construction of
localizations of small categories. Indeed, some months ago I found
that I could use Freyd's adjoint functor theorem to construct such
localizations : I exhibited an explicit solution set that solved
the problem, thus obtaining an abstract proof of existence, without
having bother with messy, though more explicit, graph constructions.
I was quite happy with this argument, but recently I realized that
I was kind of doing twice the same job : that's because I had previously
proved the cocompleteness of the category Cat of small categories
again via Freyd's adjoint theorem; and now I've just noticed that
once one knows that Cat is cocomplete, one can construct arbitrary
localizations by a sequence of suitable push-outs in Cat! This gives
a rather more conceptual and transparent approach. The idea
is borrowed from homotopy theory : indeed, there is an analogous
technique for inverting given 1-simplices in simplicial sets, that
can be transposed essentially verbatim to small categories, and
that's how I now construct localizations of small categories in
my notes!
July 10, 2023 : today I've completed the
construction of the Joyal model category structure on the category of
simplicial sets. From the construction, it is clear that every ∞-category
is a fibrant object for this category, but more work is needed to show the
converse : that will be my next goal! Another interesting result
that is now included in my notes is the construction of the canonical
model category structure on the category Cat of small categories;
I also show that this is the unique model category structure on
Cat whose weak equivalences are the equivalences of categories : I
follow a proof that I found on the website
nLab, an
impressive online collaborative math resource.
July 28, 2023 : today's version's biggest change
is that the long chapter 3 has split into two separate chapters : the new
chapter 3 now contains only the basic theory of model categories and derived
functors, and it is followed by a chapter 4 that is dedicated to
the construction of model categories, especially via the small
object argument, since the model categories that we need are
typically cofibrantly generated. Another important recent change
is the addition of a section in chapter 2, dedicated to augmented
simplicial sets, joins and slices : this material finds important
applications in our study of left and right fibrations in section 5.5.
And finally, I've corrected a very minor mistake in the proof of
theorem 4.3.2. For today, that's all folks!
August 9, 2023 : many news in today's draft : the
first important result is Joyal's theorem that identifies Kan complexes with
the ∞-groupoids. Next, I've added a rather technical result, that
morally says that every termwise invertible natural transformation
between ∞-categories is actually invertible. This is an
important ingredient in the proof of another of Joyal's theorems,
that is the latest addition in these notes, and that shows that
the ∞-categories are precisely the fibrant objects of the
Joyal model category structure; also, the fibrations between
∞-categories are precisely the isofibrations, just as in the
canonical model structure on Cat (the category of small categories).
October 6, 2023 : today I'm finally
done with the homotopy theory of ∞-categories. The text now
consists of 6 chapters : the last two are dedicated respectively
to the Kan-Quillen model category structure on simplicial sets,
and to the Joyal model category structure on the same category.
Chapter 6 also contains a discussion of higher homotopy groups
and of Serre's long exact sequence for a Kan fibration, since
this material is needed for the study of ∞-categories,
especially for characterizing weak categorical equivalences,
fully faithful functors and essentially surjective functors
of ∞-categories. All of this corresponds roughly to the first
3 chapters of Cisinski's lecture notes. I will now take a longish
break from writing : I plan to re-read all that I've written
so far, and then to add more background foundational material
on set theory; after which, I'll of course resume learning
and writing about ∞-categories and homotopical algebra.
November 1, 2023 : Last week I've begun
the long and slow process of reading all the text that I've written
so far. Today I can report that I've finished reading the first 20
pages : I've corrected many small embarrassing misprints, but nothing
of note.
It will take a while, so you won't see much happening in the next
several weeks, but I'll post regular updates.
November 17, 2023 : Today I've finished reading
the first 42 pages, i.e. up to the end of section 1.6. I've corrected many
little typos, and I've made some small additions to the text, such as
extra details for some proofs, but nothing important. Maybe the most
memorable change is that I've extended the construction of what I call
"global colimit" to arbitrary indexing categories (whereas earlier I
only dealt with filtered indexing categories); this is a device that
provides some language for dealing with cases where the colimit of a
given set-valued functor doesn't quite exist, due to set-theoretical
issues, but one has a class that enjoys the universal property of such
a colimit (the point being that in our set-theoretic framework, a "real"
colimit would have to be a set, rather than just a class). Anyway, this
is one of those nit-picking issues that appear very infrequently, and
are of potential interest only to the most dedicated set-theoreticians,
I guess, but if you really want to know more, check out example 1.1.19.
December 1, 2023 : Today I've finished reading
the first 62 pages. As usual, I've corrected many small misprints and a
few minor mistakes, but nothing important to report.
December 17, 2023 : Today I've finished reading
the first 80 pages. As usual, I've corrected many small misprints and a
few minor mistakes, but nothing much to report. The only somewhat notable
change is that I've modified axiom (EZ0) in the definition of Eilenberg-Zilber
categories : then the remark after the definition can be suppressed. Also, I
added some details and corrected some minor mistakes in the proof of proposition
1.10.12, that provides the basics of the calculus of fractions for categories.
December 28, 2023 : Today I've finished reading
the first 103 pages. The most notable change is in the proof of proposition
2.5.15, which now has a few more details; also, paragraph 2.5.14 has been
heavily revised, and this required adding a couple of remarks in earlier
sections : remarks 2.3.5 and 1.5.11. Also, the last part of the proof of
proposition 2.5.11 needed some corrections.
January 1, 2024 : Today I've finished reading
the first 120 pages. The most notable revisions are in the proof of proposition
3.1.9(v), which was slightly incomplete. Also, there was a slight mistake in
condition (b) of definition 3.1.16, which caused some confusion in the proof
of lemma 3.1.18.
February 11, 2024 : I have been silent for a
while, but work has been proceeding apace, though in unexpected directions.
The starting point for this diversion was an observation that I remembered
reading in Quillen's "Homotopical Algebra" : he proves that one can also
obtain (up to equivalence) the homotopy category H_C of a model category C
by inverting weak equivalences inside the subcategory C_c of fibrant objects
(or alternatively C_f of cofibrant objects) of C. I had already included
this result, but then Quillen also remarked that the localization of weak
equivalences in C_c can be done in two steps : first one forms a new
category C'_c with the same objects as C_c, and whose morphisms are the
homotopy classes of morphisms of C_c; next, one inverts in C'_c the homotopy
classes of weak equivalences (and similarly if one starts from C_f).
The advantage of this two-step procedure, is that the homotopy classes
of weak equivalences in C_c admit a calculus of fractions, which allows
to give a more explicit description of the morphisms of the homotopy
category of C. This last remark (which generalizes what one does in order
to construct the derived category of an abelian category...) is the one
that I wanted to include as well. Now, Quillen justifies his remark by
proving that the localization functor C'_c-->H_C has a fully faithful adjoint,
and then he quotes a result from Gabriel-Zisman's book. So I checked the
latter, and included the relevant propositions to complete Quilllen's
argument. But I also found in Gabriel-Zisman's book some further interesting
result about localizations of additive or abelian categories, so I included
that as well. But there was still something to be desired, since Gabriel and
Zisman are not so explicit about their set-theoretical foundations, and
in general, localizations tend to give large objects. Especially, a question
arises which is relevant to Quillen's discussion : suppose that one knows
that for a given category C and given class of morphisms S, the localization
C_S of C that inverts S exists; suppose moreover that one knows that S
admits a calculus of fractions. Then, one can show that the morphisms
X-->Y of C_S are represented by equivalence classes [X,Y] of "fractions
with denominators in S" in the expected way ; that is, there is an obvious
map [X,Y]-->Hom_{C_S}(X,Y) that is surjective. By the general theory of
calculus of fractions, one knows moreover that, if the class [X,Y] is
a set, then this map is bijective. Now the question is : assuming that
the localization C_S exists, and that S admits a calculus of fraction, is the
above map always a bijection? That is, do these condition imply that
[X,Y] is a set? Unexpectedly (at least, for me) the answer turns out to
be negative, in general : I asked Gabber, and he found a counterexample!
In order to properly deal with all this material, it is convenient to add
some more foundational material, which mostly amounts to introducing
a suitable language : especially, it is useful to have a notion of what
I call "wide categories" that are exactly like categories, except that
the morphisms between any two given objects are only classes (rather
than sets); then of course most standard elementary category theory
extends to the wide context, with some care, because one needs to make
sure that the operations for handling such large classes are legitimate...
So, this is basically what I'm doing now, and I'm advancing slowly but
steadily. The takeaway is that, when quoting results about the existence
of localizations in situations where one inverts a class of morphism that
admits calculus of fractions, one should take care to ensure that the
classes [X,Y] of "fractions with denominators in S" are really sets,
even in cases where one knows a priori that the localization does exist,
because otherwise one only knows by general nonsense that [X,Y] maps
surjectively onto the set of morphisms X-->Y in the localization, but
one does not has necessarily the bijectivity of these maps.
Is that clear? Raise your hand if you stayed with me till here!
And for tonight, that's all folks!
March 3, 2024 : That's it : the long digression
on localization of categories is done with! At least, for the time being :
the first chapter will certainly undergo more sysmic changes in the future.
If you are curious, Gabber's counterexample that I mentioned in the previous
post can be found starting on page 80 : it encompasses proposition 1.2.14
and example 1.2.15. An interesting feature of this counterexample is that
it uses some constructions from combinatorial group theory known as
HNN extensions! So there you go : an application of group theory to pure
category theory. Quite neat, isn't it? Now we can resume the reading of
the notes : stay tuned!
March 29, 2024 : Today I'm finally done with
the reading of the first 158 pages, that is, up to the end of section 3.4,
dedicated to derived functors, in the context of model categories. It took
more time than expected, because halfway through I realized that some
intermediate constructions could be improved : earlier on I had introduced
fibrant-cofibrant replacements for objects of a model category; this is
a standard tool that is used notably for constructing the homotopy
category of a model category : one defines it first on the subcategory
of fibrant-cofibrant objects (where no localization is needed : one only
takes homotopy classes of morphisms), and then one projects the whole model
category onto the homotopy category of the fibrant-cofibrant objects, via
fibrant-cofibrant replacements (this generalizes what one does when
constructing the derived category of an abelian category : there the
fibrant-cofibrant replacements are given by injective or projective
resolutions, depending on the context). But while re-reading this material
I realized that there are two more similar projections from the model
category, respectively onto the subcategory of fibrant objects, and onto
the subcategory of cofibrant objects (more precisely, since my fibrant
and cofibrant replacements are not functorial on the nose, I project
on the homotopy category of fibrant - or cofibrant - objects). These
two constructions are more basic than the fibrant-cofibrant one : indeed,
the latter can be obtained by combining the former ones. So, I had to do
many small changes, that taken together amount to quite a considerable
modification of the text. This has also impacted the section on derived
functors, since left derived functors are more directly related to
cofibrant replacements, whereas right derived functors employ fibrant
replacements.
April 17, 2024 : Today I've finished reading
the first 174 pages, that is, up to the end of chapter 3 : hurray!
Again, it took somewhat longer than I expected, largely due to teaching
distractions. The main changes are in the statement of proposition 3.5.11,
which had a small mistake (without any consequence), and in the proof of
proposition 3.6.5, which is now slightly shorter, because parts of it have
been extracted and dealt with separately, with more care and more details :
these parts concern auxiliary results about fibrant and cofibrant objects
in slice categories X/C, when C is already a model category, and the
induced model category on X/C is induced by C, as in proposition 3.2.4(iii);
then we also consider the functor x_!:X'/C-->X/C induced by a morphism
x:X-->X' : since C is finitely cocomplete, x_! has a left adjoint x^!,
and the pair (x^!,x_!) is a Quillen adjunction. All these complements
are now spread between remark 3.2.5 and examples 3.3.4, 3.4.6(iii,iv)
and 3.4.19. Henceforth I should be able to advance more quickly, since
teaching is over until next fall.
April 26, 2024 : Today I've finished reading the
first 190 pages. The most notable correction concerns the definition of k-small
categories, k-small object in a cocomplete category, and k-accessible functor
(for any cardinal k) : namely, in all these definition there appeared a strict
inequality (of the form : "something is strict less than k"), which conflicted
with several other references, in which the same definitions have instead
a non-strict inequality. The difference is mostly harmless, except that when
one has strict inequalities, certain assertions are valid only for regular
cardinals k, whereas with non-strict inequalities they are valid for arbitrary
infinite cardinals. However, in at least one place, the difference becomes
rather annoying : namely, in Proposition 4.2.11(ii), which requires a one-line
proof in case one has non-strict inequalities, whereas it would require some
non-trivial identities concerning exponentials of cardinals, if one adopted
the strict inequalities in the definition of k-accessible functors. This is
not such a trivial issue, since -while investigating the problem- I've learned
that the properties of cardinal exponentiation are strongly dependent on the
chosen set-theoretical framework : in particular, on whether or not one
accepts the generalized continuous hypothesis, and I definitely do not wish
to have a proposition in my notes to depend on any extraneous set-theoretical
assumptions. So, I switched throughout to the more standard definitions
with non-strict inequalities : this has involved going back to page one and
checking every instance where these notions appear, but luckily I've been able
to do that rather quickly.