CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. March 21: Chang-Yeon Chough (Adjoint functor theorem)
(1) statement of the theorem
(2) adjoint functors preserve limits /colimits
(3) accessible categories
(4) presentable categories
(5) compact objects and Freyd’s theorem
Adjoint functors are everywhere. If you pick up a book you’ll see adjoint functors.
These are useful in your real life. I’ll give you some criterion for when you can find
adjoint functors. My game plan, let me say one more thing, I’ll say everything in
terms of 1-categories, but if you replace every category with ∞-categories and sets
with spaces, some model of spaces, then everything will be the same except on a
set of measure zero.
I’ll state the theorem without any explanation.
Theorem 1.1 (Adjoint functor theorem). Let F ∶ C → D be a functor between
presentable categories (I’ll explain this later).
(1) F admits a right adjoint if and only if F preserves all colimits
(2) F admits a left adjoint if F is accessible and preserves all limits
Let me give you a quick application. Gabriel talked about the free group functor.
You can construct a free Abelian group on a set, but you can think of the existence
of the functor as a consequence of this theorem. The forgetful functor will preserve
all limits, and then by the adjoint functor theorem it will admit a left adjoint. If a
left adjoint exists, it’s essentially unique by the Yoneda lemma. Theoretically, you
have a free Abelian functor. Concretely if you’d like to construct the free Abelian
group, you can write down a formula. That concrete description is never useful.
The definition in your first year as a grad student, the construction is, you have a
word, juxtaposition, blah blah blah, you only ever use the universal property of a
free group, you never use the construction.
If you’re an algebraic geometer, given a Grothendieck topology you can describe
a sheaf, and there’s a forgetful functor from sheaves to presheaves, that satisfies
this theorem and admits a left adjoint called sheafification.
That was the statement of the theorem. Second, this is one of the most important
consequences of having adjoint functors. As always, one direction of this theorem
is immediate, which is that if F admits a right adjoint, it will preserve all colimits,
and if F admits a left adjoint, it will preserve all limits.
Proposition 1.1. Let L ∶ C ⇄ D ∶ R be an adjunction. There are many different
notations for adjunctions, this one means that L is left adjoint to R and L goes
from C to D and R goes from D to C. Then L preserves all colimits which exist in
C and R preserves all limits that exist in D.
1
2 GABRIEL C. DRUMMOND-COLE
The proof, what does this mean, you have a family of objects indexed by some
category (c
i
), with morphisms and compatibilities and so on, so a functor p from I
to C, I assume nothing on the index category. Then we can talk about the colimit
of p, and that’s an object in C. I can apply L and get an object in D. So when I
say that L preserves colimits, I can instead take L to my diagram and then have
a diagram in D and I can talk about the colimit of L ○ p. So we’re comparing
L(colim c
i
) and colim(Lc
i
). You have a map from the latter to the former and
I claim this is an isomorphism in D. I’ll give a formal proof that I like, if you
don’t like this argument, close your eyes for thirty seconds. I’ll implicitly use the
Yoneda lemma. To say that these are isomorphic, it’s enough to show that maps
into an arbitrary guy d are isomorphic. Choose any object D and think about
Hom
D
(L(colim c
i
), D) and by adjunction
Hom
D
(L(colim c
i
), D) ≅ Hom
C
(colim c
i
, RD)
which is the same thing as lim Hom
C
(c
i
, RD). Apply adjunction again, by natural-
ity this is
lim Hom
C
(c
i
, RD) ≅ lim Hom
D
(Lc
i
, D).
and then what happens, this is by the universal property of colimits, isomorphic to
Hom
D
(colim Lc
i
, D). This is true for every D and so this means they are isomor-
phic.
What does it mean to have colim Lc
i
? You have a map L(c
i
) → L(c
j
), all
compatible with the colimit. If you have any object compatible with the maps on
the top, there could be many morphisms, then you get a filler.
L(c
i
)
//
%%
L(c
j
)
yy
colim L(c
i
)
D
now you forget the middle thing and think about adjunction. That amounts by
adjunction, the outer guy, to exactly this data:
c
i
//
##
c
j
{{
colim c
i
RD
and when you apply the adjunction to this the middle guy is L(colim c
i
) which has
the unique lift. So then you have the same thing, they satisfy the same universal
property. That’s the easiest proof, not rigorous. For example, one application,
with forgetful functors, let’s say from vector spaces, R-modules, to sets. That guy
satisfies this condition so it admits a left adjoint and is a right adjoint. Think
about the Hom tensor adjunction, if you think about the Hom functor that admits
a left adjoint, which is the tensor product of modules. That tensor product is a
DERIVED SEMINAR 3
left adjoint, and therefore it preserves all colimits. If you go back to first year grad
studies, when you study commutative algebra, if you take direct limit or colimit of
M
i
and tensor with N , that’s the colimit of (M
i
⊗ N ). You do this by hand for
direct limits, but this is a formal property now because tensoring with N is a left
adjoint.
Let me skip accessibility. An accessible category, this arises naturally in mathe-
matics. Sets, Abelian groups, modules, are accessible, there are examples of inter-
est. In real life, like for Abelian groups, you may face set theoretical issues. Then
accessiblity allows you to avoid set theoretical difficulties.
Let me give you an idea for how this goes, I want to assume I preserve all colimits
and show that F admits a right adjoint, so that Hom
D
(F c, d) ≅ Hom
C
(c, Gd) for
some G. Having a right adjoint means I want to have such an object, if F admits
this adjoint, I should have an object Gd satisfying this condition. So at this point
we don’t know, but we want to fill in the blank and define Gd. What does that
mean, that means
