CHANGING TYPES
DAVID I. SPIVAK
1. Notation
Let F : A → C and G : B → C be functors. We denote by (F ↓ G) the category
whose objects are pairs (a, b, f ) where a ∈ Ob(A), b ∈ Ob(b) and f : F (a) → G(b)
is a morphism in C, and whose morphism sets are given by
Hom(F ↓G) ((a, b, f ), (a0 , b0 , f 0 )) = {(x : a → a0 , y : b → b0 )|yf = f 0 x}.
If F : A → Cat is a functor and B is a category, we sometimes write (A ↓ {B})
to denote (F ↓ iB ), where iB : {∗} → Cat is the category B.
If C = Cat we introduce a new category (F ⇓ G) whose objects are the same as
those in (F ↓ G) but for which morphisms sets are given by
Hom(F ⇓G) ((a, b, f ), (a0 , b0 , f 0 )) = {(x, y, α)|x : a → a0 , y : b → b0 , α : yf → f 0 x}.
We write (F ⇑ G) to denote the category with the direction of the natural transformation reversed: same objects, but morphism sets given by
Hom(F ⇑G) ((a, b, f ), (a0 , b0 , f 0 )) = {(x, y, α)|x : a → a0 , y : b → b0 , α : f 0 x → yf }.
Again, we may write (F ⇓ {B}) to denote (F ⇓ iB ). Explicitly, the objects in
f
f
(F ⇓ {B}) are functors F (a) −
→ B, where a ∈ A, and the morphisms (F (a) −
→
f0
B) −→ (F (a0 ) −→ B) are pairs (x, α) giving a natural transformation diagram
x
/ F (a0 )
F (a)
CC
α
CC
zz
CC =⇒ zzz0
CC
z
f
! |zz f
B
Given F : A → Cat, let F̄ : A → Cat denote the functor a 7→ F (a)op . Clearly,
(F ↓ {B}) is isomorphic to (F̄ ↓ {B op }). The category (F̄ ⇓ {B op }) can be
identified with the category whose objects are functors F (a) → B, where a ∈ A,
f0
f
and the morphisms (F (a) −
→ B) −→ (F (a0 ) −→ B) are pairs (x, α) giving a natural
transformation diagram
/ F (a0 ) .
F (a)
CC
z
α
CC
z
CC ⇐= zzz0
C
z
f
C! |zz f
B
x
Clearly, this is the category (F ⇑ {B}) = (F̄ ⇓ {B op }).
This project was supported in part by the Office of Naval Research.
1
2
DAVID I. SPIVAK
2. Grothendieck construction
Let T : S → Cat be a functor. The Grothendieck construction for T is given by
Gr(T ) := ({∗} ⇓ T ).
Explicitly, Gr(T ) is the category with objects (S, s) where S ∈ Ob(S) and s ∈ T (S).
The morphism sets in Gr(T ) are given by
Hom((S, s), (S 0 , s0 )) = {(y : S → S 0 , α : T (y)(s) → s0 )}.
There is a natural projection Gr(T ) → S given by (S, s) 7→ S. It is a split
fibration; that is, for any y : S → S 0 and s ∈ S, there is a canonical choice of map
(S, s) → (S 0 , T (y)(s)) lying over y.
Lemma 2.0.1. Let σ : C → S and T : S → Cat be functors. The diagram
/ Gr(T )
Gr(T σ)
y
C
/S
is a pullback square.
Recall that a pullback square as in Lemma 2.0.1 defines a functor from the
sections of Gr(T ) → S to the sections of Gr(T σ) → C.
Lemma 2.0.2. Suppose given two functors f, g : S → Cat; let Nat(f, g) denote
the set of natural transformations from f to g. There is a natural isomorphism
∼
=
HomS (Gr(f ), Gr(g)) −
→ Nat(f.g).
t
Definition 2.0.3. There is a functor Γ : (Cat ⇓ {Cat}) → Cat defined as follows.
On objects, Γ(f : C → Cat) is defined to be the global sections of the functor
π : Gr(f ) → C, i.e. the category whose objects are functors s : C → Gr(f ) with
π ◦ s = idC , and whose morphisms are natural transformations s → s0 .
To define Γ on morphisms, suppose we are given a natural transformation diagram
/D.
C CC
z
CC
z
α
CC ⇐= zzz
z g
f CC
! }zz
Cat
x
By Lemma 2.0.1, there is a canonical functor Γ(g) → Γ(gx). By Lemma 2.0.2,
there is a canonical functor Γ(gx) → Γ(f ) induced by α. The composition gives
the desired functor.
Lemma 2.0.4. Suppose that σi : Ci → Cat, for i = 1, 2, 3, are three functors and
suppose that f : C2 → C1 and g : C2 → C3 are functors too. There is an induced
σ4
functor C4 := C1 qC2 C3 −→
Cat. Taking Grothendieck constructions yeilds the
CHANGING TYPES
3
diagram
Gr(σ1 ) VV
VVVV
q8
q
VVVV
q
qq
VV+
Gr(σ2 ) VV
Gr(σ4 )
VVVV
qq8
VVVV
q
q
q
VV+
g0
Gr(σ3 )
f0
C
1 WWWWW
7
f ppp
WWWWW
WWWWW pppp
W+
C2 WWWWW
7 C4
WWWWW
p
pp
p
W
p
W
W
p
g
WWW+
p
C3
All of the vertical squares are pullback squares by Lemma 2.0.1.
The natural map
Γ(σ4 ) −→ Γ(σ1 ) ×Γ(σ2 ) Γ(σ3 )
is an isomorphism.
3. Monad
Lemma 3.0.5. Suppose that C is a category and T is a monad on C. If C ∈ C is
a T -algebra, then T lifts to a monad on the category (C ↓ C).
Proof. Given f : X → C we define T (f ) to be the composition T (X) → T (C) → C.
It satisfies the axioms for a monad.
Remark 3.0.6. The material below emphasizes the role of Cat, but everything said
(Lemma 3.0.7 and 3.0.8) also works for Sets. In particular, replace the category
FC with the category Fin of finite sets, and replace the category Cat with the
category Sets.
Lemma 3.0.7. Let FC denote the category of finite categories. Consider the
endofunctor d := (FC ⇓ (−)) : Cat → Cat. There are natural transformations
η : idCat → d and µ : d2 → d, which gives d the structure of a monad on Cat.
Proof. For C ∈ Cat, define ηC : C → dC to be the functor given on objects by
c
c 7→ ({∗} →
− C), and given on morphisms by










id{∗}

0
/ {∗} 
(f : c → c ) 7→  {∗}
.
AA


}
f
AA
}


AA =⇒ }}}


0
c
A
}


A ~}} c
C
This is clearly a natural transformation.
Definining µ is more difficult. Let C ∈ Cat be a category; we will define µC .
Let X be a finite category and let σ : X → (FC ⇓ C) be an object in d2 C. We
need to establish some notation for σ. On objects x ∈ X, denote σ(x) by
σx : Sx → C,
4
DAVID I. SPIVAK
where Sx is a finite category. On morphisms f : x → y in X, denote σ(f ) by






σf

/ Sy
(σf , σf] ) :  Sx @

@
]
~
@
σf

@@ =⇒
~~

~~ σy
σx @@@

~
~
C





.




Define µ(X, σ) ∈ Cat as the category with objects
Ob(µ(X, σ)) := {(x, sx )|x ∈ X, sx ∈ Sx }
and morphisms
Homµ(X,σ) ((x, sx ), (y, sy )) = {(f, f ] )|f : x → y, (f ] : σf (sx ) → sy ) ∈ Sy }.
Note that since X and each Sx is finite, so is µ(X, σ).
There is a canonical functor µ(X, σ) → C given on objects by (x, sx ) 7→ σx (sx )
and given on morphisms by sending (f, f ] ) : (x, sx ) → (y, sy ) to the composite
σf]
σy (f ] )
σx (sx ) −→ σy σf (sx ) −−−−→ σy (sy ).
Thus µ(X, σ) is indeed in (FC ⇓ C.
So far, we have only established µ on objects. On morphisms
m
/ X0 ,
X II
u
II
u
]
u
m
II
uu
=⇒
uu σ0
σ III
u
$
zu
(FC ⇓ C)
define µ(m, m] ) : µ(X, σ) → µ(X 0 , σ 0 ) as the functor given as follows. Send the
object (x, sx ) ∈ µ(X, σ) to (m(x), m]x (sx )) ∈ µ(X 0 , σ 0 ). Send (f, f ] ) : (x, sx ) →
(y, sy ) to
0
(m(x), (m]y ◦ f ] : σmf
(m]x (sx )) → m]y (sy ))) : µ(X, σ) → µ(X 0 σ 0 ).
There is an obvious natural transformation
/ µ(X 0 , σ 0 )
µ(X, σ)
GG
GG
vv
GG =⇒
vv
v
GG
v
G# {vvv
C
because m] : σ 0 ◦ m → σ was assumed natural.
CHANGING TYPES
5
We draw the following diagram, in case it is useful to the reader:
Sx
σf
/
AA
AA σy
AA
⇓ AA 0
/C
/ Smy
0
=
σmy
m]y
m]x
0
Smx
σx
σf]
Sy ⇐
0
σmx
]
σmf
⇑
0
σmx
We forgo showing that this really is a monad.
Lemma 3.0.8. The global sections functor Γ : (FC ⇓ Cat) → Cat gives Cat the
structure of a d-algebra.
Therefore, one can consider d as a monad on (Cat ⇓ {Cat}).
Proof. The second assertion follows from the first by Lemma 3.0.5.