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 definitions 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.
Mai 6, 2024 : Today I've finished reading the
first 213 pages, that is, until the end of section 4.5. I've corrected many
small misprints, but nothing especially remarkable.
May 19, 2024 : Today I've finished reading
the first 236 pages. The first notable change concerns proposition 5.1.14,
which was not stated correctly (the proof is unchanged). Similarly,
I've corrected and expanded a little lemma 5.2.1. Next, I've
(hopefully) improved the discussion of the first properties of Kan's
Sd and Ex functors (at the beginning of section 5.3), and the proof
of lemma 5.3.4. Several other misprints and minor slips have been also
corrected, as usual.
May 24, 2024 : Today I've finished reading
that first 245 pages, that is, up to the end of chapter 5 : olé!
The most notable change is the correction of a subtle mistake in
the proof of theorem 5.3.11 : it took me one day to figure out that
my proof had a gap, one more day to realize that the gap was an
actual mistake (rather than just an incomplete or missing argument),
and then one more day to find a fix and write it up.
May 29, 2024 : Today I've finished reading
that first 260 pages, that is, up to the end of section 6.3. The most
notable changes concern :
*) corollary 6.1.10, that has been slightly improved and included as
part (ii) of corollary 6.1.9
*) remark 6.2.2(ii) that seemed to be slightly incorrect, so I've
introduced a notion of "strictly conservative morphism" to fix this,
but I don't expect that I will make much use of this notion
*) and especially, proposition 6.3.3, where I was unable to understand
my previous proof, so I have redone it, in the style of proposition
5.1.10 : indeed, the same argument appears in the context of anodyne
extensions, inner anodyne extensions, and left/right anodyne extensions.
June 10, 2024 : Today I've finished reading
that first 276 pages, that is, up to the end of section 6.5. I've made
a large number of modifications to the text : I've corrected many small
(but embarrassing!) mistakes, and I've added many small (but cute!)
improvements. There are too many to detail them all, but I'll just
mention two : a new part (iv) of corollary 6.5.5 now states that the
adjoint pair given by the functors k(-,X) and h(-,X) is a Quillen
adjunction, for every ∞-category X; and a new part (ii) of
proposition 6.2.6 states that the adjoint pair given by products
and internal homs (in the category of simplicial sets) is a Quillen
adjunction, relative to the Joyal model structure : the same assertion
was already observed earlier, for the Kan-Quillen model structure
(corollary 5.1.12(i)), but the Joyal variant is useful in other situations.
June 18, 2024 : Today I've finished reading
that first 291 pages, that is, up to the construction of the Serre long
exact sequence for a Kan fibration. I had to correct some mistakes in
my presentation : I had stated that the loop functor on pointed
∞-categories preserves all representable limits, but my argument
only proves that it preserves small products, as well as certain fibre
products : the latter are important in the construction of the Serre
exact sequence (see example 6.6.5(iii)). By the way, I do not know whether
the category of ∞-categories is complete : it has small products,
but I don't know whether it has equalizers. Even if it had equalizers,
they might not commute with the inclusion into the larger category of
all simplicials sets, so that would not imply left exactness of the
loop functor. For my construction of Serre exact sequence, I follow
Cisinski's presentation in his lecture notes on ∞-categories,
which makes use of some special properties of Kan fibrations and Kan
complexes; on the other hand, I believe that there is an abstract
approach to loop functors that makes sense in arbitrary model categories,
so that one has a version of Serre long exact sequence in any such
model category. In particular, there should be an assertion valid
more generally for isofibrations (instead of Kan fibrations), obtained
by applying this abstract approach to Joyal's model category structure
on simplicial sets... Maybe at some point I'll try to learn all this.
June 24, 2024 : Today I've finished
reading the whole text : hurray! I've concluded with some house-cleaning :
I've rearranged somewhat the contents of sections 1.6 and 1.7.
Now I'll take a short break, since I've got some exams to grade;
then I'll be back and start again adding new material : first I'm
going to add some background on set-theoretical foundations, including
cardinal and ordinal numbers. Next, we'll get back to ∞-categories :
stay tuned!
July 12, 2024 : And we're back! As announced
earlier, the next goal is to include set-theoretic foundational material.
For this, I am following Mendelson's book on Mathematical Logic. Now, in
order to do justice to such a subject, one must include a certain amount
of preliminaries concerning formal languages and axiomatic systems; hence,
I am starting from the very beginning, that is, from propositional calculus
and its axiomatisation, which I have just finished adding today : the
first section of the notes contains this material, up to a proof of the
completeness and consistency of the axiomatic theory of propositional
logic. Concerning the proof of consistency : I feel that Mendelson's
discussion leaves some questions unaddressed, of a meta-logical nature :
essentially, I fret about what exactly is achieved, given that such a
proof must be carried out within an axiomatic system which is of necessity
stronger than propositional logic itself, so it seems to be tainted by
circularity... I attempt a discussion of these questions in the final
remark of this section (remark 1.1.22) but I am not entirely certain
that I am doing a good job at it, so if you think you have a better
answer, or if you can point to a reference which gives a more
thorough/thoughtful answer, please let me know by all means!
July 20, 2024 : Today's version adds the
first elements of first order logic and elementary model theory : this
material is contained in a new section 1.2 which is complete, and a
section 1.3 that I've just started, so watch it grow!
I am following Mendelson's textbook very closely : some parts are
basically cut-and-pasted from it. So, not so much to report right now,
and we'll be toiling with first order logic and model theory for a while,
I expect.