Journal of Homotopy and Related Structures, vol. 1(1), 2006, pp.47–78
PSEUDO-CATEGORIES
N. MARTINS-FERREIRA
(communicated by Ross Street)
Abstract
We provide a complete description of the category of pseudocategories (including pseudo-functors, natural and pseudonatural transformations and pseudo modifications). A pseudocategory is a non strict version of an internal category. It was
called a weak category and weak double category in some earlier papers. When internal to Cat it is at the same time a
generalization of a bicategory and a double category. The category of pseudo-categories is a kind of “tetracategory” and it
turns out to be cartesian closed in a suitable sense.
1.
Introduction
The notion of pseudo-category1 considered in this paper is closely related and
essentially is a special case of several higher categorical structures studied for example by Grandis and Paré [8], Leinster [3], Street [11],[12], among several others.
We have arrived to the present definition of pseudo-category (which some authors
would probably call a pseudo double category) while describing internal bicategories in Ab [5]. We even found it easier, for our particular purposes, to work with
pseudo-categories than to work with bicategories. Defining a pseudo-category we begin with a 2-category, take the definition of an internal category there, and replace
the equalities in the associativity and identity axioms by the existence of suitable
isomorphisms which then have to satisfy some coherence conditions. That is, let C
be a 2-category, a pseudo-category in (internal to) C is a system
(C0 , C1 , d, c, e, m, α, λ, ρ)
where C0 , C1 are objects of C,
d, c : C1 −→ C0 , e : C0 −→ C1 , m : C1 ×C0 C1 −→ C1
The author thanks to Professor G. Janelidze for reading and suggesting useful changes to clarify
the subject, and also to Professors R. Brown and T. Porter for the enlightning discutions during
the seminars presented at Bangor.
Received April 14, 2005, revised April 11, 2006; published on May 5, 2006.
2000 Mathematics Subject Classification: Primary 18D05, 18D15; Secondary 57D99.
Key words and phrases: Bicategory, double category, weak category, pseudo-category, pseudo double category, pseudo-functor, pseudo-natural transformation, pseudo-modification, weakly cartesian closed.
c 2006, N. Martins-Ferreira. Permission to copy for private use granted.
°
1 In
the previous work [6] the word ”weak” was used with the same meaning. We claim that
”pseudo” is more apropriate because it is the intermediate term between precategory and internal
category. Also it agrees with the notion of pseudo-functor, already well established.
Journal of Homotopy and Related Structures, vol. 1(1), 2006
48
are morphisms of C , with C1 ×C0 C1 the object in the pullback diagram
π
C1 ×C0 C1 −−−2−→


π1 y
C1

c
y
d
C1
;
−−−−→ C0
α : m (1C1 ×C0 m) −→ m (m ×C0 1C1 ) ,
λ : m hec, 1C1 i −→ 1C1 , ρ : m h1C1 , edi −→ 1C1 ,
are 2-cells of C (which are isomorphisms), the following conditions are satisfied
de = 1c0 = ce,
(1.1)
dm = dπ2 , cm = cπ1 ,
(1.2)
d ◦ λ = 1d = d ◦ ρ,
c ◦ λ = 1c = c ◦ ρ,
(1.3)
d ◦ α = 1dπ3 , c ◦ α = 1cπ1 ,
(1.4)
λ ◦ e = ρ ◦ e,
(1.5)
and the following diagrams commute
m◦(1C1 ×C0 α)
/•
•
00
00
±± ±
00
±
±
α◦(1C1 ×C0 1C1 ×C0 m) ±
00α◦(1C1 ×C0 m×C0 1C1 )
±±
00
00
±± ±
0»
±§±
•B
•
|
BB
|
BB
||
BB
||
|
BB
||
B
α◦(m×C0 1C1 ×C0 1C1 ) BB
|| m◦(α×C0 1C1 )
|
BB
BB |||
à ~|
•
α◦(1
×
<ec,1
>)
C1 C0
C1
/•
•?
??
Ä
Ä
??
ÄÄ
??
ÄÄ
??
Ä
ÄÄ m◦(ρ×C0 1C1 )
m◦(1C1 ×C0 λ) ??
??
ÄÄ
Ä
? ÄÄÄ
•
(1.6)
(1.7)
Journal of Homotopy and Related Structures, vol. 1(1), 2006
49
Examples:
1. When C=Set with the discrete 2-category structure (only identity 2-cells) one
obtains the definition of an ordinary category since α, λ, ρ are all identities;
2. When C=Set with the codiscrete 2-category structure (exactly one 2-cell for
each pair of morphisms) one obtain the definition of a precategory since α, λ, ρ
always exist and the coherence conditions are trivially satisfied;
(This result applies equally to any category)
3. When C=Grp considered as a 2-category: every group is a (one object) category and the inclusion functor
Grp −→ Cat
induces a 2-category structure in Grp, where a 2-cell
τ : f −→ g , (f, g : A −→ B group homomorphisms)
is an element τ ∈ B, such that for every x ∈ A,
g (x) = τ f (x) τ −1 .
With this setting, a pseudo-category in Grp is described (see [7]) by a group
homomorphism
∂ : X −→ B,
an arbitrary element
δ ∈ ker ∂
and an action of B in X (denoted by b · x for b ∈ B and x ∈ X) satisfying
∂ (b · x) = b∂ (x) b−1
∂ (x) · x0 = x + x0 − x
for every b ∈ B, x, x0 ∈ X. Note that the difference to a crossed module
(description of an internal category in Grp) is that in a crossed module the
element δ = 1.
The pseudo-category so obtained is as follows: objects are the elements of
B, arrows are pairs (x, b) : b −→ ∂x + b and the composition of (x0 , ∂x + b) :
∂x+b −→ ∂x0 +∂x+b with (x, b) : b −→ ∂x+b is the pair (x0 + x − δ + b · δ, b) :
b −→ ∂x0 + ∂x + b. The isomorphism between (0, ∂x + b) ◦ (x, b) = (x, b) ◦ (0, b)
and (x, b) is the element (δ, 0) ∈ X o B . Associativity is satisfied, since
(x00 , ∂x0 + ∂x + b) ◦ ((x0 , ∂x + b) ◦ (x, b)) = ((x00 , ∂x0 + ∂x + b) ◦ (x0 , ∂x + b)) ◦
(x, b) .
4. When C=Mor(Ab) the 2-category of morphisms of abelian groups, the above
definition gives a structure which is completely determined by a commutative
Journal of Homotopy and Related Structures, vol. 1(1), 2006
50
square
∂
A1 −−−−→


k1 y
A0

k
y 0
∂0
B1 −−−−→ B0
together with three morphisms
λ, ρ : A0 −→ A1 ,
η : B0 −→ A1 ,
satisfying conditions
k1 λ = 0 = k1 ρ,
k1 η = 0,
and it may be viewed as a structure with objects, vertical arrows, horizontal
arrows and squares, in the following way (see [6], p. 409, for more details)
(b,x)
−
−−−−−−−−−−−
→
b
¡ b ¢

d y
µ
b + ∂ 0 (d)
b x
d y
b + k0 (x)
¡
 b+k0 (x)¢
y d+k1 (y) ,
¶
∗
−−−−−−−−−−−−−→
(b+∂ 0 (d),x+∂(y))
where ∗ stands for b + ∂ 0 (d) + k0 (x + ∂ (y)) = b + k0 (x) + ∂ 0 (d + k1 (y)) .
5. When C=Top (with homotopy classes as 2-cells) we find the following particular example. Let X be a space and consider the following diagram
d
−−→
e
X ×X X −→ X ←−
−−→X
I
I
m
I
c
I
where X is equipped with the compact open topology and X I ×X X I with
the product topology (I is the unit interval), with
X I ×X X I = {hg, f i | f (0) = g (1)}
and d, e, c, m defined as follows
d (f ) = f (0)
c (f ) = f (1)
ex (t) = x
½
g (2t) , t < 12
m (f, g) =
f (2t − 1) , t >
1
2
with f, g : I −→ X (continuous maps) and x ∈ X. The homotopies α, λ, ρ are
the usual ones.
Journal of Homotopy and Related Structures, vol. 1(1), 2006
51
6. When C=Cat the objects C0 and C1 are (small) categories, and the morphisms
d, c, e, m are functors. We denote the objects of C0 by the first capital letters
in the alphabet (possible with primes) A, A0 , B, B 0 , ... and the morphisms by
first small letters in the alphabet a : A −→ A0 , b : B −→ B 0 , ... . We will
denote the objects of C1 by small letters as f, f 0 , g, g 0 , ... and the morphisms
by small greek letters as ϕ : f −→ f 0 , γ : g −→ g 0 , ... . We will also consider
that the functors d and c are defined as follows
C1
C0
d%
a : A −→ A0
ϕ : f −→ f 0
c&
b : B −→ B 0
hence, the objects of C1 are arrows f : A −→ B, f 0 : A0 −→ B 0 , that we
will always represent using inplace notation as A f / B , A0 f 0 / B 0 to
distinguish from the morphisms of C0 , and thus the morphisms of C1 are of
the form
A
²
A0
f
a
®¶
ϕ
f0
/B
²b .
/ B0
The functor e sends a : A −→ A0 to
A
²
A0
/A
idA
a
®¶
id
ida
A0
²a ,
/ A0
while the functor m sends hγ, ϕi to γ ⊗ ϕ as displayed in the diagram below
A
a
²
A0
f
®¶
ϕ
f0
/B
²b
/ B0
/C
² c 7−→
/ C0
g
®¶
γ
g0
A
a
²
A0
Each component of α is of the form
A
1A
²
A
h⊗(g⊗f )
®¶
αh,g,f
(h⊗g)⊗f
/D
² 1D ,
/D
g⊗f
®¶
γ⊗ϕ
g 0 ⊗f 0
/C
²c .
/ C0
Journal of Homotopy and Related Structures, vol. 1(1), 2006
52
while the components of λ and ρ are given by
A
1A
²
A
/B
² 1B
/B
idB ⊗f
®¶
λf
f
A
1A
²
A
,
/B
² 1B .
/B
f ⊗idA
®¶
ρf
f
Thus, a description of pseudo-category in Cat is as follows.
-
A pseudo-category in Cat is a structure with
objects: A, A0 , A00 , B, B 0 , ...
morphisms: a : A −→ A0 , a0 : A0 −→ A00 , b : B −→ B 0 , ...
pseudo-morphisms: A f / B , A0 f 0 / B 0 , B g / C , ...
and cells:
A
²
A0
a
f
ϕ
®¶ 0
f
/B
²b
/ B0
,
A0
a ²
A00
f0
0
f
®¶
ϕ0
00
/ B0
0
²b
/ B 00
,
B
²
B0
b
/C
g
®¶ 0
g
γ
² c , ...
/ C0
where objects and morphisms form a category
a00 (a0 a) = (a00 a0 ) a,
1A0 a = a1A ;
pseudo-morphisms and cells also form a category
ϕ00 (ϕ0 ϕ) = (ϕ00 ϕ0 ) ϕ,
1f 0 ϕ = ϕ1f ,
with 1f being the cell
A
²
A
1A
f
®¶
1f
f
/B
² 1B ;
/B
for each pair of pseudo-composable cells γ, ϕ, there is a pseudo-composition γ ⊗ ϕ
A
a
²
A0
g⊗f
®¶
γ⊗ϕ
g 0 ⊗f 0
/C
²c ;
/ C0
satisfying
(γ 0 γ) ⊗ (ϕ0 ϕ) = (γ 0 ⊗ ϕ0 ) (γ ⊗ ϕ) ,
1g⊗f = 1g ⊗ 1f ;
(1.8)
Journal of Homotopy and Related Structures, vol. 1(1), 2006
53
for each morphism a : A −→ A0 , there is a pseudo-identity ida
A
²
A0
/A
idA
a
®¶
²a
/ A0
ida
idA0
satisfying
id1A = 1idA
ida0 a = ida0 ida ;
there is a special cell αh,g,f for each triple of composable pseudo-morphisms h, g, f
A
h⊗(g⊗f )
1A
²
A
αh,g,f
®¶
(h⊗g)⊗f
/D
² 1D ,
/D
natural in each component, i.e., the following diagram of cells
h ⊗ (g ⊗ f )


η⊗(γ⊗ϕ)y
αh,g,f
−−−−→
(h ⊗ g) ⊗ f

(η⊗γ)⊗ϕ
y
αh0 ,g0 ,f 0
h0 ⊗ (g 0 ⊗ f 0 ) −−−−−→ (h0 ⊗ g 0 ) ⊗ f 0
commutes for every triple of pseudo-composable cells ϕ, γ, η
A
a
²
A0
/B
f
ϕ
®¶
g
²b
/ B0
f0
®¶
g0
γ
/C
²c
/ C0
h
®¶
η
h0
/D
²d ;
/ D0
to each pseudo-morphism f : A −→ B there are two special cells
A
²
A
/B
² 1B
/B
idB ⊗f
1A
®¶
λf
f
,
A
²
A
1A
f ⊗idA
®¶
ρf
f
/B
² 1B ,
/B
natural in f , that is, to every cell ϕ as above, the following diagrams of cells commute
λf
idB ⊗ f −−−−→


id1B ⊗ϕy
id
B0
0
λf 0
f

ϕ
y
⊗ f −−−−→ f
0
ρf
,
f ⊗ idA −−−−→


ϕ⊗id1A y
λf 0
f

ϕ
y
f 0 ⊗ idB 0 −−−−→ f 0
.
Journal of Homotopy and Related Structures, vol. 1(1), 2006
54
And furthermore, the following conditions are satisfied whenever the compositions
are defined
f ⊗ (g ⊗ (h ⊗ k))
f ⊗αg,h,k
vv
αf,g,h⊗k vv
v
v
v
vv
zvv
(f ⊗ g) ⊗ (h ⊗ k)
/ f ⊗ ((g ⊗ h) ⊗ k)
HH
HH α
HHf,g⊗h,k
HH
HH
H$
(f ⊗ (g ⊗ h)) ⊗ k
TTTT
TTTT
TTT
αf ⊗g,h,k TTTT
TTT
)
jj
jjjj
j
j
j
j
jjjjαf,g,h ⊗k
ju jjj
((f ⊗ g) ⊗ h) ⊗ k
f ⊗ (1 ⊗ g)
αf,1,g
HH
HH
HH
HH
HH
f ⊗λg HHH
HH
HH
$
/ (f ⊗ 1) ⊗ g
v
.
vv
vv
v
v
v
vv
vv ρf ⊗g
v
vv
vz v
f ⊗g
Examples of pseudo-categories internal to Cat include the usual bicategories of
Spans, Bimodules, homotopies, ... where in each case it is also allowed to consider
the natural morphisms between the objects in order to obtain a vertical categorical
structure. For example in the case of spans we would have sets as objects, maps as
morphisms, spans A ←− S −→ B as pseudo-morphisms and the cells being triples
(h, k, l) with the following two squares commutative
A ←−−−− S −−−−→




hy
ky
B


yl
A0 ←−−−− S 0 −−−−→ B
.
A pseudo-category in Cat has the following structures: a category (with objects
and morphisms); a category (with pseudo-morphisms and cells); a bicategory (considering only the morphisms that are identities); a double category (if all the special
cells are identity cells).
Other examples as Cat (with modules as pseudo-morphisms) may be found in
[3] or [8].
The present description of pseudo double category (internal pseudo-category in
Cat) is the same given by Leinster [3] and differs from the one considered by Grandis
and Paré [8] in the sense that they also have
idA = idA ⊗ idA .
In the following sections we will provide a complete description of pseudo-functors,
natural and pseudo-natural transformations and pseudo-modifications. We prove
that all the compositions are well defined (except for the horizontal composition
Journal of Homotopy and Related Structures, vol. 1(1), 2006
55
of pseudo-natural transformations which is only defined up to an isomorphism). In
the end we show that the category of pseudo-categories (internal to some ambient
2-category C) is Cartesian closed up to isomorphism. We will give all the definitions in terms of the internal structure to some ambient 2-category and also explain
what is obtained in the case where the ambient 2-category is Cat. While doing some
proofs we will make use of Yoneda embedding and consider the diagrams in Cat
rather than in the abstract ambient 2-category.
We will also freely use known definitions and results from [4],[1],[2] and [10].
2.
Pseudo-Functors
Let C be a 2-category and suppose
C = (C0 , C1 , d, c, e, m, α, λ, ρ) ,
C 0 = (C00 , C10 , d0 , c0 , e0 , m0 , α0 , λ0 , ρ0 )
(2.1)
are two pseudo-categories in C.
A pseudo-functor F : C −→ C 0 is a system
F = (F0 , F1 , µ, ε)
where F0 : C0 −→
C00 , F1
: C1 −→ C10 are morphisms of C,
µ : F1 m −→ m0 (F1 ×F0 F1 ) , ε : F1 e −→ e0 F0 ,
are 2-cells of C (that are isomorphisms2 ), the following conditions are satisfied
d0 F1 = F0 d,
c0 F1 = F0 c,
(2.2)
d0 ◦ µ = 1F0 dπ2 ,
c0 ◦ µ = 1F0 cπ1 ,
(2.3)
d 0 ◦ ε = 1 F0 ,
c0 ◦ ε = 1F0 ,
(2.4)
and the following diagrams commute
µ(1×C0 m)
ÄÄ
ÄÄ
Ä
ÄÄ Ä
•@
@@
@@
@@
µ(m×C0 1)
Â
F1 α
2 Some
/•
@@ 0
@@m (1F1 ×µ)
,
@@
@Â
•
ÄÄ
Ä
ÄÄ 0
ÄÄÄ α (F1 ×F0 F1 ×F0 F1 )
/•
•
m0 (µ×1F1 )
•
(2.5)
authors (example Grandis and Paré in [8, 9]) consider the notion of pseudo - which corresponds to the present one - but also consider the notions of lax and colax where the 2-cells may
not be isomorphisms.
Journal of Homotopy and Related Structures, vol. 1(1), 2006
F1 ρ
•


µh1,ediy
−−−−→
•
x
 0
ρ F1
,
m0 (1F1 ×ε)
56
(2.6)
• −−−−−−−→ •
F λ
−−−1−→
•


µhec,1iy
•
x
 0
λ F1
.
m0 (ε×1F1 )
• −−−−−−−→ •
Consider the particular case of C=Cat. Let
A
²
A0
f
a
®¶
ϕ
f0
/B
²b
/ B0
be a cell in the pseudo-category C. A pseudo-functor F : C −→ C 0 , consists of four
maps (sending objects to objects, morphisms to morphisms, pseudo-morphisms to
pseudo-morphisms and cells to cells - that we will denote only by F to keep notation
simple)
FA
Fa
²
F A0
Ff
Fϕ
®¶
F f0
/ FB ;
² Fb
/ F B0
a special cell µf,g
FA
1
²
FA
F (g⊗f )
®¶
µf,g
F g⊗F f
/ FC
²1
/ FC
to each pair of composable pseudo-morphisms f, g; a special cell εA
FA
1
²
FA
F (idA )
®¶
²A
idF A
/ FA
²1
/ FA
Journal of Homotopy and Related Structures, vol. 1(1), 2006
57
to each object A, and satisfying the commutativity of the following diagrams
F (αf,g,h )
F (f ⊗ (g ⊗ h))
/ F ((f ⊗ g) ⊗ h)
µf,g⊗h
,
µf ⊗g,h
²
F (f ) ⊗ F (g ⊗ h)
²
F (f ⊗ g) ⊗ F (h)
F (f )⊗µg,h
µf,g ⊗F (h)
²
F (f ) ⊗ (F (g) ⊗ F (h)) 0
²
/ (F (f ) ⊗ F (g)) ⊗ F (h)
αF f,F g,F h
F (f ⊗ idA )

µf,idA 
y
F (ρf )
−−−−→
F (f )
x
ρ0
 Ff
,
F (f )⊗εA
F (f ) ⊗ F (idA ) −−−−−−→ F (f ) ⊗ idF (A)
F (idB ⊗ f )

µidB ,f 
y
F (λf )
−−−−→
F (f )
x
λ0
 Ff
,
εB ⊗F (f )
F (idB ) ⊗ F (f ) −−−−−−→ idF (B) ⊗ F (f )
whenever the pseudo-compositions are defined.
Return to the general case.
Let F : C −→ C 0 and G : C 0 −→ C 00 be pseudo-functors in a 2-category C.
Consider C and C 0 as in (2.1) and let
C 00 = (C000 , C100 , d00 , c00 , e00 , m00 , α00 , λ00 , ρ00 ) ,
¡
¢
F = F0 , F1 , µF , εF ,
¡
¢
G = G0 , G1 , µG , εG .
The composition of the pseudo-functors F and G is defined by the formula
¡
¡
¢ ¡
¢ ¡
¢ ¡
¢¢
GF = G0 F0 , G1 F1 , µG ◦ (F1 ×F0 F1 ) · G1 ◦ µF , εG ◦ F0 · G1 ◦ εF
(2.7)
where ◦ represents the horizontal composition in C and · represents the vertical
Journal of Homotopy and Related Structures, vol. 1(1), 2006
58
composition, as displayed in the diagram below
m
e
m0
e0
m00
e00
C1 ×C0 C1 −−−−→ C1 ←−−−− C0






F1 × F 0 F1 y
µF ⇓ F1 y
εF ⇓ yF0
C10 ×C00 C10 −−−−→ C10 ←−−−− C00
 .





G 1 × G 0 G1 y
εG ⇓ yG0
µG ⇓ G1 y
C100 ×C000 C100 −−−−→ C100 ←−−−− C000
Proposition 1. The above formula to compose pseudo-functors is well defined.
Proof. Consider the system
¢
¡
GF = G0 F0 , G1 F1 , µGF , εGF
with µGF , εGF as in (2.7). We will show that GF is a pseudo-functor from the
pseudo-category C to the pseudo-category C 00 .
It is clear that G0 F0 : C0 −→ C000 , G1 F1 : C1 −→ C100 , are morphisms of the ambient 2-category C and µGF : G1 F1 m −→ m00 (G1 F1 ×G0 F0 G1 F1 ) , εGF : G1 F1 e −→
e00 G0 F0 are 2-cells of C and they are isomorphisms.
Conditions (2.2) are satisfied and
¡¡
¢ ¡
¢¢
d00 µGF = d µG ◦ (F1 ×F0 F1 ) · G1 ◦ µF
¡
¢ ¡
¢
= d ◦ µG ◦ (F1 ×F0 F1 ) · d ◦ G1 ◦ µF
¡
¢ ¡
¢
= 1G0 d0 π20 ◦ (F1 ×F0 F1 ) · G0 ◦ d0 ◦ µF
³
´
= 1G0 d0 π0 (F1 ×F F1 ) · (G0 ◦ 1F0 dπ2 )
2
0
= 1G0 d0 F1 π2 · 1G0 F0 dπ2
= 1G0 F0 dπ2 ,
as well c00 µGF = 1G0 F0 cπ1 , hence (2.3) holds. Also
¡¡
¢ ¡
¢¢
d00 εGF = d00 εG ◦ F0 · G1 ◦ εF
¡
¢ ¡
¢
= d00 ◦ εG ◦ F0 · d00 G1 ◦ εF
¡
¢
= (1G0 ◦ F0 ) · G0 d0 ◦ εF
= 1G0 F0 · (G0 ◦ 1F0 )
= 1G0 F0 · 1G0 F0 = 1G0 F0 ,
and similarly c00 εGF = 1G0 F0 , so conditions (2.4) are satisfied.
Commutativity of diagrams (2.5) , (2.6) follows from Yoneda Lemma and the
Journal of Homotopy and Related Structures, vol. 1(1), 2006
59
commutativity of the following diagrams
H(αf,g,h )
Hf ⊗(g⊗h)
¢¢
¢¢
¢
²
¢¢
µH
f,g⊗h ¢
¢
GFf ⊗Fg⊗h
¢
II
¢¢
pp
II
¢¢ pppp
II
¢
p
II
¢
p
I$ G(α0F f,F g,F h )
¡¢¢wppp
/ (2)
Hf ⊗ Hg⊗h
(1)
==NNN
uu
u
== NNN
uu
== NNN
uu
NN'
==
uu
u
z
==
==Ff ⊗ GFg ⊗Fh
Hf ⊗µH
==
g,h
==
==
Á ²
α00
Hf,Hg,Hh
Hf ⊗ (Hg ⊗ Hh )
B ,f
HidB
H(λf )
/ Hf
QQQ
O
QQQ
(
/ GidF ⊗Ff
GFidB ⊗Ff
B
|
|
λ00
Hf
|
²
|
|
|
G
⊗
H
|
idFB
f
||
RRR
ff3
RRR
fffff
||
f
f
f
|
f
R(
² ~| ffff
/ idHB ⊗ Hf
⊗ Hf
H
HidB ⊗f
µH
id
/ H(f ⊗g)⊗h)
==
==
==
== µH
²
== f ⊗g,h
GFf ⊗g ⊗Fh
NNN ===
u
NNN ==
uu
NNN ==
uu
u
u
NN' =Á
zuu
H
f ⊗g ⊗ Hh
II
p¢¢
p
II
p
II
pp ¢
II
ppp ¢¢
I$
wppp ¢¢¢
GFf ⊗Fg ⊗ Hh ¢¢
¢ µH ⊗H
¢¢ f,g h
¢
¢¢
² ¡¢¢
/ (Hf ⊗ Hg ) ⊗ Hh
²B ⊗Hf
H(ρ )
f
/ Hf
PPP
O
PPP
(
/ GFf ⊗idF
GFf ⊗FidA
A
|
|
µH
ρ00
f,idA
Hf
|
²
||
|
H
⊗
G
|
f
id
FA
||
QQQ
fff3
QQQ
||
fffff
f
f
|
f
Q(
f
² ~| fff
/ Ff ⊗ idHA
Hf ⊗ HidA
H
Hf ⊗idA
Hf ⊗²A
where (1) = GFf ⊗(Fg ⊗Fh ) and (2) = G(Ff ⊗Fg )⊗Fh . We also use the abbreviations
H = GF and Ff or F f instead of F (f ) to save space in the diagram.
Composition of pseudo-functors is associative and there is an identity pseudofunctor for every pseudo-category, namely the pseudo-functor
1C = (1C0 , 1C1 , 1m , 1e )
for the pseudo-category
C = (C0 , C1 , d, c, e, m, α, λ, ρ) .
Journal of Homotopy and Related Structures, vol. 1(1), 2006
60
Given a 2-category C, we define the category PsCat(C) consisting of all pseudocategories and pseudo-functors internal to C.
3.
Natural and pseudo-natural transformations
Let C be a 2-category and suppose
C = (C0 , C1 , d, c, e, m, α, λ, ρ) ,
C 0 = (C00 , C10 , d0 , c0 , e0 , m0 , α0 , λ0 , ρ0 )
(3.1)
are pseudo-categories in C and
¡
¢
F = F0 , F1 , µF , εF ,
¡
¢
G = G0 , G1 , µG , εG
(3.2)
are pseudo-functors from C to C 0 .
A natural transformation θ : F −→ G is a pair θ = (θ0 , θ1 ) of 2-cells of C
θ0 : F0 −→ G0
θ1 : F1 −→ G1
satisfying
d0 ◦ θ1 = θ0 ◦ d
c0 ◦ θ1 = θ0 ◦ c
and the commutativity of the following diagrams of 2-cells
•


µF y
θ ◦m
−−1−−→
•

 G
yµ
m0 ◦(θ1 ×θ0 θ1 )
• −−−−−−−−−→ •
θ ◦e
1
• −−−
−→


εF y
e0 ◦θ0
•

G
yε
.
• −−−−→ •
A pseudo-natural transformation T : F −→ G is a pair
T = (t, τ )
where t : C0 −→ C10 is a morphism of C,
τ : m0 hG1 , tdi −→ m0 htc, F1 i
Journal of Homotopy and Related Structures, vol. 1(1), 2006
61
is a 2-cell (that is an isomorphism); the following conditions are satisfied
d0 t = F0
c0 t = G0
(3.3)
d0 ◦ τ = 1d0 F1
c0 ◦ τ = 1c0 G1
(3.4)
and the following diagrams of 2-cells are commutative3
α−1 (G1 ×G0 G1 ×G0 t)
Ä
ÄÄ
Ä
Ä
ÄÄ Ä
m0 hµ−1
G ,tdπ2 i
•
/•
•
τm
²
•@
@@
@@
@@
0
m htcπ1 ,µF i
Â
²
•
/•
•
(3.5)
@@ 0
@@m hG1 π1 ,τ π2 i
@@
@Â
•
,
α(G1 ×G0 t×F0 F1 )
ÄÄ
ÄÄ 0
Ä
ÄÄÄ m hτ π1 ,F1 π2 i
α(t×F0 F1 ×F0 F1 )
τe
/•
•
@@ 0
Ä
@@m h1t ,εF i .
λ t ÄÄ
@@
Ä
Ä
@Â
Ä
ÄÄ
•
• OOO
o
OOO
ooo
o
OOO
o
o
O
ooo 0
m0 hεG ,1t i OOO
O' woooo ρ t
•
(3.6)
0
In the case C=Cat: let W, W 0 be two pseudo-categories in Cat, and F, G : W −→
W two pseudo-functors. Given a cell
0
A
²
A0
f
a
®¶
ϕ
f0
/B
²b
/ B0
in W , we will write
FA
²
F A0
Fa
3G
Ff
®¶
Fϕ
F f0
/ FB
Fb
0
²
/ FB
and
GA
²
GA0
Ga
×G0 t ×F0 F1 : C1 ×C0 C0 ×C0 C1 −→ C1 ×C0 C1 ×C0 C1
t ×F0 F1 ×F0 F1 : C0 ×C0 C1 ×C0 C1 −→ C1 ×C0 C1 ×C0 C1
G1 ×G0 G1 ×G0 t : C1 ×C0 C1 ×C0 C0 −→ C1 ×C0 C1 ×C0 C1
1
Gf
®¶
Gϕ
Gf 0
/ GB
² Gb
/ GB 0
Journal of Homotopy and Related Structures, vol. 1(1), 2006
62
for the image of ϕ under F and G.
The description of natural and pseudo-natural transformations in this particular
case is as follows:
- While a natural transformation θ : F −→ G is a family of cells
FA
²
GA
θA
Ff
®¶
θf
Gf
/ FB
² θB ,
/ GB
one for each pseudo-morphism f in W , such that for every cell ϕ in W , the square
θf
F f −−−−→


F ϕy
Gf

Gϕ
y
θf 0
F f 0 −−−−→ Gf 0
is commutative as displayed in the picture below
Fa
/ NN
NNN
FfN
NN
N
N
b NNN θ
² N NNNN F ϕ ®¶ 0 NNNNθNf
²F
NNBN
/
NNNN NNN
NNN θA NNN F f NN
N
N
N
NNN
NNN
NNNθB0 NNNN
NNNθf 0 N "*
&/
&
N
NN
NNN
NNN Gf
N
θA0 NNN
N
N
N
Gϕ
Ga
NN& ²
NN& ²
"* ®¶
/
Gf 0
;
Gb
and furthermore, given two composable pseudo-morphisms g, f and an object A in
W , the following squares are commutative
µF
g,f
F (g ⊗ f ) −−−−→ F g ⊗ F f



θ ⊗θ
θg⊗f y
yg f
µG
g,f
G (g ⊗ f ) −−−−→ Gg ⊗ Gf
εF
F (idA ) −−−A−→ idF A



id
θidA y
y θA .
G (idA ) −−−−→ idGA
εG
A
- Rather a pseudo-natural transformation T : F −→ G consists of two families of
Journal of Homotopy and Related Structures, vol. 1(1), 2006
63
cells
FA
Fa ²
F A0
tA
®¶
ta
t A0
/ GA
² Ga
/ GA0
and
FA
1
²
FA
Gf ⊗tA
®¶
τf
tB ⊗F f
/ GB
²1
/ GB
with a a morphism and f a pseudo-morphism of W , as displayed in the following
picture
Fa
NNN
4</ N
Ff
NNN
ppp FNbNNNN
Fϕ
p
p
¶
®
²
²N
p
NNN tA NNN F f 0 pppppτf pp /4< NNNN tB NNN
p
NN ®¶ tb NNN
NN ®¶ ta NNN pp
p
p
&/
& p
tB 0 N
t A0 N
Gf
ppτp0
p
NNN
NNN
p
f
p
Gϕ
Ga
NN& ²
NN& ² ppp
®¶ 0
/
Gb
Gf
such that (t is a functor)
ta0 a = ta ta0
t 1A = 1 t A
(τ is natural)
τf
Gf ⊗ tA −−−−→ tB


Gϕ⊗ta y
⊗ Ff

t ⊗F ϕ
yb
τf 0
,
Gf 0 ⊗ tA0 −−−−→ tB 0 ⊗ F f 0
and for every two composable pseudo-morphisms A
f
/B
g
/ C , the following
Journal of Homotopy and Related Structures, vol. 1(1), 2006
64
diagrams of cells in W 0 are commutative
/ G (g) ⊗ (G (f ) ⊗ tA )
(G (g) ⊗ G (f )) ⊗ tA
RRR
nn
RRR
n
n
n
RRR
n
nn
RRR
n
n
R)
vnn
G (g ⊗ f ) ⊗ tA
G (g) ⊗ (tB ⊗ F (f ))
²
²
tC ⊗ F (g ⊗ f )
(G (g) ⊗ tB ) ⊗ F (f )
PPP
PPP
PPP
PPP
(
ll
lll
l
l
lll
lu ll
/ (tC ⊗ F (g)) ⊗ F (f )
tC ⊗ (F (g) ⊗ F (f ))
G (idA ) ⊗ tA
oo
o
o
oo
o
o
ow oo
idGA ⊗ tA V
VVVV
VVVV
VVVV
VVVV
VVVV
VV+
/ tA ⊗ F (idA )
OOO
.
OOO
OOO
OO'
tA ⊗ idF A
hhh
h
h
h
hh
hhhh
hhhh
h
h
h
hhh
tA hs
Return to the general case.
Let C be a 2-category and suppose C, C 0 , C 00 are pseudo-categories in C and
F, G, H : C −→ C 0 , F 0 , G0 : C 0 −→ C 00 are pseudo-functors. Natural transformations θ, θ0 , θ̇
F
C
θ
G
®¶
®¶
θ̇
H
"
/ C0
<
F0
®¶
θ0
*
4 C 00
G0
may be composed horizontally with θ0 ◦ θ = (θ00 , θ10 ) ◦ (θ0 , θ1 ) = (θ00 ◦ θ0 , θ10 ◦ θ1 )
obtained
composition
of 2-cells of C, and vertically with θ̇ · θ =
³
´ from the horizontal
³
´
θ̇0 , θ̇1 ·(θ0 , θ1 ) = θ̇0 · θ0 , θ̇1 · θ1 obtained from the vertical composition of 2-cells
of C. Clearly both compositions are well defined, are associative, have identities
and satisfy the middle interchange law. This fact may be stated as in the following
theorem.
Theorem 1. Let C be a 2-category. The category PsCat(C) (with pseudo-categories,
pseudo-functors and natural transformations) is a 2-category.
Composition of pseudo-natural transformations is much more delicate.
Again let C be a 2-category and suppose C, C 0 are pseudo-categories in C,
F, G, H : C −→ C 0 are pseudo-functors (as above) and consider the pseudo-natural
transformations
T
S
F −→ G −→ H
Journal of Homotopy and Related Structures, vol. 1(1), 2006
65
with
T = (t, τ ) ,
S = (s, σ) .
Vertical composition of pseudo-natural transformations S and T is defined as
S ⊗ T = (m0 hs, ti , σ ⊗ τ )
(3.7)
where
σ ⊗ τ = α hsc, tc, F1 i · m0 h1sc , τ i · α−1 hsc, G1 , tdi · m0 hσ, 1td i · α hH1 , sd, tdi . (3.8)
The above formula in the case C=Cat is expressed as follows
(s ⊗ t)a = sa ⊗ ta ,
FA
Fa ²
F A0
tA
®¶
ta
t A0
/ GA
² Ga
/ GA0
/ HA
² Ha ;
/ HA0
sA
sa
sA®¶ 0
and
(σ ⊗ τ )f = α (sB ⊗ τf ) α−1 (σf ⊗ tA ) α,
as displayed in the following picture
(στ )f
Hf ⊗ (sA ⊗ tA ) −−−−→ (sB ⊗ tB ) ⊗ F f

x

α
αy

(Hf ⊗ sA ) ⊗ tA


σf ⊗tA y
sB ⊗ (tB ⊗ F f )
x
s ⊗τ
B f
.
(sB ⊗ Gf ) ⊗ tA −−−−1
−→ sB ⊗ (Gf ⊗ tA )
α
Return to the general case.
Theorem 2. The vertical composition of pseudo-natural transformations is well
defined.
Proof. Consider C, C 0 as in (3.1), F, G as in (3.2) , H = (H0 , H1 , µH , εH ) and
S, T as above. Clearly (st) = m0 hs, ti : C0 −→ C10 is a morphism of C and στ :
m0 hH1 , (st) di −→ m0 h(st) c, F1 i is a 2-cell of C that is an isomorphism (is defined
as a composition of isomorphisms).
Conditions (3.3) and (3.4) are satisfied
d0 m0 hs, ti = d0 π20 hs, ti
= d0 t
= F0 ,
Journal of Homotopy and Related Structures, vol. 1(1), 2006
66
also c0 m0 hs, ti = c0 s = H0 , and
d0 ◦ (σ ⊗ τ ) = d0 ◦ α hsc, tc, F1 i · m0 h1sc , τ i · α−1 hsc, G1 , tdi · m0 hσ, 1td i · α hH1 , sd, tdi
= d0 ◦ α hsc, tc, F1 i · d0 ◦ m0 h1sc , τ i ·
d0 ◦ α−1 hsc, G1 , tdi · d0 ◦ m0 hσ, 1td i · d0 ◦ α hH1 , sd, tdi
= 1d0 F1 · 1d0 F 1 · 1d0 td · 1d0 td · 1d0 td
= 1d0 F1 · 1d0 td = 1d0 F1 · 1F0 d = 1d0 F1 · 1d0 F1 = 1d0 F1
with similar computations for c0 ◦ (σ ⊗ τ ) = 1c0 H1 .
Commutativity of diagrams (3.5) and (3.6) is obtained using Yoneda Lemma,
writing the respective diagrams and adding all the possible arrows to fill them in
order to obtain the following mask and diamond
?
ÄÄ ???
ÄÄÄ
Â
??
?? ÄÄÄ
 ÄÄ
²
/ ?
??
?Â
²
Ä ????
Ä
Ä
Â
/ ÄÄ?
?? ÄÄ
? ÄÄÄ
²
/²?
?
?
?
Ä
?? ÄÄÄ ???
Ä
² ÄOÄ
 ÄÄ
Â
??
OOO
Ä
?
Ä
OOO
? ÄÄÄ
OOO
OOO
OO' ²
??
??
Â
/? ?
ÄÄ ???
Ä
Â
/ ?Ä
?? ÄÄ?
? ÄÄ
(mask)
/
Ä
ÄÄ
Ä
Ä
/
²
²
Ä ????
Ä
Ä
Ä
Ä
ÄÄÄ
Â
/ ÄÄ
??
?? ÄÄÄ
 ÄÄ
²o ²
_??
Ä ???? ÄÄ
??
Ä
ÄÄÄ
 ÄÄÄ
²
??
o
o
o
?? ÄÄÄ
ooo
 ÄÄ
ooo
o
o
oo
/ ² woo
/
­ ///
²G 444
­
²
­ //
² 44
44
­­
²²
º O
­
7
44
­
OOO
o //
o
­
²
o
O
o
½
­¥ O
//
'
/ /o
DDOOO ²²²
o
/º ooozz7
//
²
DDO' ¨²
²
o
z
DD ??
//
²
Ä zzz
DD ? ¨²²
/ º ÄÄÄzzz
DD?? 4
DD?? 44
­­ ÄzÄzz
DD?? 4
­­ÄzÄzÄz
DD??44
­
DD??44 ­­ ÄzÄz
DD?4 ­ÄzÄz
"Â ½ ¥­ÄÄ|z
(diamond)
in which squares commute by naturality, hexagons commute by definition of (σ ⊗ τ ),
octagons commute because S, T are pseudo-natural transformations, pentagons in
the diamond commute by the same reason and all the other pentagons and triangles
commute by coherence.
The horizontal composition of pseudo-natural transformations is only defined up
to an isomorphism and it will be considered at the end of this paper.
In the next section we define square pseudo-modification ( simply called pseudomodification) and show that given two pseudo-categories C, C 0 , we obtain a pseudocategory by considering the pseudo-functors as objects, natural transformations
Journal of Homotopy and Related Structures, vol. 1(1), 2006
67
as morphisms, pseudo-natural transformations as pseudo-morphisms and pseudomodifications as cells. So, in particular, we will show that the vertical composition of
pseudo-natural transformations is associative and has identities up to isomorphism.
We also show that PsCat is Cartesian closed up to isomorphism, that is, instead of
an isomorphism of categories PsCat(A × B, C)∼
=PsCat(A,PsCAT(B, C)) we get an
equivalence of categories PsCat(A × B, C)∼PsCat(A,PsCAT(B, C)).
4.
Pseudo-modifications
Let C be a 2-category. Suppose C, C 0 are pseudo-categories in C, F, G, H, K :
C −→ C 0 are pseudo-functors, T = (t, τ ) : F −→ G, T 0 = (t0 , τ 0 ) : H −→ K are
pseudo-natural transformations and θ = (θ0 , θ1 ) : F −→ H, θ0 = (θ00 , θ10 ) : G −→ K
are two natural transformations.
A pseudo-modification Φ (that will be represented as)
F
²
H
T
Φ
θ
®¶
T0
/G
0
²θ
/K
is a 2-cell of C
Φ : t −→ t0
satisfying
d0 ◦ Φ = θ0
c0 ◦ Φ = θ00
(4.1)
and the commutativity of the square
τ
• −−−−→


m0 hθ10 ,Φ◦diy
•

 0
ym hΦ◦c,θ1 i
.
τ0
• −−−−→ •
(4.2)
Consider the case where C=Cat. Suppose W, W 0 are two pseudo-categories in
Cat, F, G, H, K : W −→ W 0 are pseudo-functors, T : F −→ G, T 0 : H −→ K
are pseudo-natural transformations and θ : F −→ G, θ0 : H −→ K are natural
transformations.
A pseudo-modification Φ
F
θ
²
H
T
®¶
Φ
T0
/G
0
²θ
/K
Journal of Homotopy and Related Structures, vol. 1(1), 2006
68
is a family of cells
FA
θA
²
HA
/ GA
0
² θA
/ KA
tA
®¶
ΦA
t0A
of W 0 , for each object A in W , where the square
Φ
tA −−−A−→


ta y
Φ
t0A

t0
ya
,
(4.3)
0
tA0 −−−A−→ t0A0
commutes for every morphism a : A −→ A0 in W (naturality of Φ) and the square
τf
Gf ⊗ tA −−−−→ tB


θf0 ⊗ΦA y
⊗ Ff

Φ ⊗θ
y B f
,
(4.4)
Kf ⊗ t0A −−−0−→ t0B ⊗ Hf
τf
commutes for every pseudo-morphism f : A −→ B in W .
Both squares (4.3) and (4.4) may be displayed together with full information, for
a ϕ in W , as follows
/
Ff
s D
szzs8@ DDDD
ssss
z
ss DDD
s
s
z
s
s
s
s
z
DD
DD
s
ss
sszzz
ss
DDssssssss F ϕ
DD
sszzzzzz
ss
s
¶
®
²
²
s
s
s
s
z
0
s
/
z
s
s
s
z
t
F
f
s
D
@
8
D
A
D
z
s
s
s
s
stBD
s
s
s
ss sssDssDssDssDssssss DDsDssssss sssszzzzzzz τf szzszzszz DDDssDsssss DDD
θA sss
s
s
s
s
s
s
D
s
z
ss ss D tass
s z
ss D tb DDD
s zz
s
ss
ss ssssssssssssθssfs DsDss®¶ssss ssDDsDs" θzBzzzzz ssszzzzzz sssssss DD ®¶
ss
D"
s
s
s
s z
s
s
z
ss
s
tB 0D
ss sssssssssssssss
ssssssssstAs0DsDsDs ss sssszzzzzzτfs0 sssssssssssGf
s/
ss
s
s
D
s
s
s
s
s
s
s
s
D
s
s
z
s
s
s
s
θ A0 s
ss
sss
ssssssssss
sssssssssssss sDsD sss zzzz sssssssssss
sDs
ss
ss
sssss
ss DDDD
ss ssssssssssssssssφA ssssssssssθssfsss0 ssss ssss sDsDDs θBzz0zzzzsssssssssssφssBsss sss®¶ ssGϕ
s
ss
s
s
s
"²
ysD
ss " ² zz sssssss
ss Hfsu} sssss
ssss sss/ ys ss
ssss0 0 ssss
s/
ss
DD
sssssssssssszθzz0 8@ DDDsss ssss sssssssssssssssss
sss
sssssssssθsGf
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
z
D
DD ssss
s
s
s
s
sss
sssssss Bss
ssss sss zzz Ass D ss ss ssssssss
sHϕ
ss
ssDD
ssss ®¶ u} ssssssssssssssφzAzz0zzzssss ssssDDDssssssssssssssssssssθf0 sssssssssssφssBss0s ssss
ss
s
s
z
²
² ysss D0 u} sss
}
u
s
s
s
sss zzzz ss 0 8@/ DysD ssst0BsDsssss
Hf 0s
t AD
DD
ssssssss ss
ss
D
zz Dss
sssD
DD
sssssssss ss θB0 0 sss
ssss zzz sτs0
s
DD t0a DDsDssDsssss zzzzszzszss f zzzzzzθAs0sss DDssDssssss t0b DDsDssDssssssssssss ssss
ss
z
sD
DD ssss DD zzzss
ss
s
ss
zzzzzssss u} sssss D 0 ®¶ u} sssssssssssDsD/" θysf0 s0 s
sz
®¶ s
z
s
"
s
y
z
z
0 u}
s
z
s
tB 0 s
t A0 D
Kf
ss
zzz s0 s
sDss
DD
ss
zzszzszssτf 0
sssss DDDD
s
DD
s
z
s
s
z
Kϕ ss
zzs
s
DD
DD
ss
D" ² yssss
D" ² zyszzszzzss
®¶ u} ssss
/
Kf 0
(4.5)
Return to the general case.
Journal of Homotopy and Related Structures, vol. 1(1), 2006
69
Let C be a 2-category and consider C, C 0 two pseudo-categories in C as in (3.1).
Suppose T, T 0 , T 00 are pseudo-natural transformations between pseudo-functors from
C to C 0 : we define for
Φ0
Φ
T −→ T 0 −→ T 00
a composition Φ0 Φ as the composition of 2-cells in C, and clearly it is well defined,
is associative and has identities. Now for θ, θ0 , θ00 natural transformations between
pseudo-functors from C to C 0 , we define for
Φ
Ψ
θ −→ θ0 −→ θ00
a pseudo-composition Ψ ⊗ Φ = m0 hΨ, Φi .
Proposition 2. Let C be a 2-category and suppose Ψ, Φ are pseudo-modifications
F
θ ²
F0
/G
T
Φ
®¶ 0
T
²θ
/ G0
/H
S
0
Ψ
®¶ 0
S
²θ
/ H0
00
with F, G, H, F 0 , G0 , H 0 pseudo-functors from C to C 0 (pseudo-categories as in (3.1)),
S, T, S 0 , T 0 pseudo-natural transformations and θ, θ0 , θ00 natural transformations as
considered above.
The formula
Ψ ⊗ Φ = m0 hΨ, Φi
for pseudo-composition of pseudo-modifications is well defined.
Proof. Recall that the composition of pseudo-modifications is given by
S ⊗ T = (m0 hs, ti , (σ ⊗ τ ))
with (σ ⊗ τ ) given as in (3.8), hence
m0 hΨ, Φi : m0 hs, ti −→ m0 hs0 , t0 i
is a 2-cell of C as required.
Conditions (4.1) are satisfied,
d0 m0 ◦ hΨ, Φi = d0 π20 ◦ hΨ, Φi = d0 ◦ Φ = θ0
c0 m ◦ hΨ, Φi = c0 π10 hΨ, Φi = c0 ◦ Ψ = θ000 .
To prove commutativity of square (4.2) we use Yoneda Lemma and the following
Journal of Homotopy and Related Structures, vol. 1(1), 2006
70
diagram, obtained by adapting (4.2) to the present case and filling its interior
GG
GG
GG
GG
GG
G#
(1)
SSSS
SSSS
SS)
k5
kkkk
k
k
k
/ kk
(2)
(3)
²
kk5
k
k
k
² kkkkk
w;
w
ww
ww
w
w
² www
/
w;
ww
w
ww
ww
w
ww
(4)
(5)
/ ² SSS
SSSS
SSS) ²
GG
GG
GG
GG
GG
G# ²
/
where hexagons commute by definition of (σ ⊗ τ ) and (σ 0 ⊗ τ 0 ), squares (1) , (3) , (5)
commute by naturality of α0 while squares (2) , (4) commute because Ψ, Φ are
pseudo-modifications (satisfy (4.4)) together with the fact that pseudo-composition
(in C 0 ) satisfies the middle interchange law (1.8).
Composition of pseudo-natural transformations is not associative, however there
is a special pseudo-modification for each triple of composable pseudo-natural transformations.
Proposition 3. Let C be 2-category and suppose F, G, H, K : C −→ C 0 are pseudofunctors in C and that S = (s, σ) , T = (t, τ ) , U = (u, υ) are pseudo-natural transformations as follows
S
T
U
F −→ G −→ H −→ K.
0
The 2-cell αU,T,S
= α0 hu, t, si is a pseudo-modification
F
1
²
F
U ⊗(T ⊗S)
®¶
α0U,T ,S
(U ⊗T )⊗S
/K
²1
/K
,
and it is natural in S, T, U, in the sense that the square
U ⊗ (T ⊗ S)


ϕ⊗(γ⊗δ)y
α0 hu,t,si
−−−−−→
(U ⊗ T ) ⊗ S

(ϕ⊗γ)⊗δ
y
α0 hu0 ,t0 ,s0 i
U 0 ⊗ (T 0 ⊗ S 0 ) −−−−−−−→ (U 0 ⊗ T 0 ) ⊗ S 0
commutes for every pseudo-modification ϕ : U −→ U 0 , γ : T −→ T 0 , δ : S −→ S 0 .
Journal of Homotopy and Related Structures, vol. 1(1), 2006
71
Proof. The 2-cell α0 hu, t, si is obtained from
hu,t,si
C0 −→ C10 ×C00 C10 ×C00 C10
−−−−−−
→
⇓ α 0 C1 ,
−−−−−−→
and
U ⊗ (T ⊗ S) = (m (1 × m) hu, t, si , (υ ⊗ (τ ⊗ σ)))
(U ⊗ T ) ⊗ S = (m (m × 1) hu, t, si , ((υ ⊗ τ ) ⊗ σ)) ,
hence
α0 hu, t, si : m (1 × m) hu, t, si −→ m (m × 1) hu, t, si
is a 2-cell of C.
Conditions (4.1) are satisfied
d0 ◦ α0 ◦ hu, t, si = 1d0 π30 hu, t, si = 1d0 s = 1F0
c0 ◦ α0 ◦ hu, t, si = 1c0 π10 hu, t, si = 1c0 u = 1K0 .
Commutativity of (4.2) follows from Yoneda Lemma and the commutativity of the
following diagram
??
??
??
?Â
²
/
O
/
Ä?
Ä
ÄÄ
ÄÄ
Ä
/O
/
²
/²?
?
??
Ä
??
ÄÄ
?? ÄÄÄ
 Ä
??
??
??
?Â
??
??
??
?Â
? ?
Ä
Ä ???
Ä
??
ÄÄ
?Â
ÄÄ
/²
/²
?
ÄÄ
Ä
ÄÄ
² ÄÄ
/
/²?
??
??
??
Â/ ²
where hexagons commute because S, T, U are pseudo-natural transformations,
squares commute by naturality and pentagons by coherence.
To prove naturality we observe that
((ϕ ⊗ γ) ⊗ δ) · (α0 hu, t, si) = (m0 hm hϕ, γi , δi) · (α0 hu, t, si)
= (m0 (m0 × 1) hϕ, γ, δi) · (α0 hu, t, si)
¢
¢ ¡
¡
= 1m0 (m0 ×1) · α0 ◦ hϕ, γ, δi · 1hu,t,si
= α0 ◦ hϕ, γ, δi
Journal of Homotopy and Related Structures, vol. 1(1), 2006
72
and
(α0 hu0 , t0 , s0 i) · (ϕ ⊗ (γ ⊗ δ)) = (α0 hu0 , t0 , s0 i) · (m0 hϕ, m0 hγ, δii)
= (α0 hu0 , t0 , s0 i) · (m0 (1 × m0 ) hϕ, γ, δi)
¢ ¡
¡
¢
= α0 · 1m0 (1×m0 ) ◦ 1hu0 ,t0 ,s0 i hϕ, γ, δi
= α0 ◦ hϕ, γ, δi .
For every pseudo-functor there is a pseudo-identity pseudo-natural transformation and a pseudo-identity pseudo-modification.
Proposition 4. Consider a pseudo-functor F = (F0 , F1 , µF , εF ) : C −→ C 0 in a 2category
C ¢(with¢ C, C 0 pseudo-categories in C as in (3.1)). The pair
¡ 0
¡ 0−1
e F0 , λ ρ0 ◦ F1 is a pseudo-natural transformation in PsCat(C)
¡
¢
idF = e´F0 , λ0−1 ρ0 F1 : F −→ F,
and the 2-cell 1e0 F0 : e0 F0 −→ e0 F0 is a pseudo-modification in PsCat(C)
F
1
²
F
idF
®¶
1idF
idF
/F
²1
/F
.
Proof. Clearly e0 F0 : C0 −→ C10 is a morphism of C, and
λ0−1 ρ0 F1 : m hF1 , e0 d0 F1 i −→ m0 he0 c0 F1 , F1 i
is a 2-cell (that is an isomorphism) of C.
Conditions (3.3) and (3.4) are satisfied,
d0 e0 F0 = F0
c0 e0 F0 = F0
¡
¢
¡
¢
d0 ◦ λ0−1 ρ0 F1 = d0 ◦ λ0−1 ρ0 ◦ F1
¡
¢
= d0 λ0−1 F1 · (d0 ρ0 F1 )
= (1d0 F1 ) · (1d0 F1 )
= (1d0 F1 ) ,
¡
¢
and similarly for c0 ◦ λ0−1 ρ0 F1 = 1c0 F1 .
Commutativity of (3.5) is obtained using Yoneda Lemma and the commutativity
Journal of Homotopy and Related Structures, vol. 1(1), 2006
73
of the diagram
Fg⊗f ⊗ idF A
oo
ooo
o
o
o
ow oo
(Fg ⊗ Ff ) ⊗ idF A
II
II
II
II
I$
F
/ idF C ⊗ Fg⊗f
PPP
u:
PPP
uu
u
PPP
uu
PPP
u
'
uu
id
⊗ (F ⊗ F )
g⊗f
FC
g
f
VVVV
4
hhhh
VVVV
h
h
h
h
VVVV
hhhh
VVVV
hhhh
VVV*
h
h
²
²
²
h
h
F g ⊗ Ff o
Fg ⊗ (Ff ⊗ idF A ) o
(idF C ⊗ Fg ) ⊗ Ff ,
:
II
OOO
nn7
II
uu
OOO
nnn
II
uu
n
OOO
u
n
II
nn
uu
OO'
I$
nnn
uu
/ (Fg ⊗ idF B ) ⊗ Ff
Fg ⊗ (idF B ⊗ Ff )
while (3.6) follows in a similar way as observed in the diagram
F (idA ) ⊗ idF A
rr
rrr
r
r
rx rr
idF A ⊗ idF A
TTTT
TTTT
TTTT
TTTT
TT*
idF A
/ idF A ⊗ F (idA )
MMM
MMM
MMM
M&
idF A ⊗ idF A .
iiii
i
i
i
iii
iiii
tiiii
This proves that idF is a pseudo-natural transformation. To prove 1idF = 1e0 F0
is a pseudo-modification we note that
1e0 F0 : e0 F0 −→ e0 F0
is a 2-cell of C,
d0 ◦ 1e0 F0 = 1d0 e0 F0 = 1F0 ,
c0 ◦ 1e´F0 = 1c0 e0 F0 = 1F0 .
To prove commutativity of square (4.2) we use Yoneda Lemma and the commutativity of the following square
0
λ0−1
F f ρF f
F f ⊗ idF A −−−−−→ idF B ⊗ F f



1id ⊗1F f
1F f ⊗1idF A y
y FB
.
0
λ0−1
F f ρF f
F f ⊗ idF A −−−−−→ idF B ⊗ F f
Proposition 5. Let C be a 2-category and suppose F, G : C −→ C 0 are pseudofunctors in C.
For every pseudo-natural transformation
T = (t, τ ) : F −→ G
Journal of Homotopy and Related Structures, vol. 1(1), 2006
74
there are two special pseudo-modifications
F
²
F
/G
²1
/G
idG ⊗T
1
®¶
T
λT
F
²
F
,
/G
²1
/G
T ⊗idF
1
®¶
T
ρT
,
with λT = λ0 ◦ t, ρT = ρ0 ◦ t both natural in T .
Proof. It is clear that λ0 ◦ t : m0 ht, e0 F0 i −→ t is a 2-cell of C, and
d0 ◦ λ0 ◦ t = 1d0 t = 1F0
c0 ◦ λ0 ◦ t = 1c0 t = 1G0 .
The commutativity of square (4.2) is obtained from the commutativity of diagram
/ (idGB ⊗ tB ) ⊗ Ff
9
tt
tt
t
tt
tt
Gf ⊗ (idGA ⊗ tA )
KK
KK
KK
KK
K%
(Gf ⊗ idGA ) ⊗ tA
idGB ⊗ (tB ⊗ Ff )
NNN
9
ss
NNN
§§
ss
§
N
s
N
§
NNN
ss
§§
&
ss
§
(idGB ⊗ Gf ) ⊗ tA −/ idGB ⊗ (Gf ⊗ tA )
§§
hhh eeeeeeeeee
§§
h
h
h
§
h
h
h
§
h
eeeee
§
hhhh eeeee
² £§§ threheheheheeeeee
Gf ⊗ tA
88
88
88
88
88
88
88
88
¿ ²
/ t B ⊗ Ff
In order to prove naturality of λT consider a internal pseudo-modification
F
θ
²
H
/G
T
Φ
®¶ 0
T
²θ
/K
0
as defined in (4.1); then, on the one hand we have
¡
¢
Φ · (λ0 ◦ t) = 1C10 ◦ Φ · (λ0 ◦ 1t )
¡
¢
= 1C10 · λ0 ◦ (Φ · 1t )
= λ0 ◦ Φ
and on the other hand we have
¡ ­
®
¢
(λ0 ◦ t0 ) · (m0 he0 θ00 , Φi) = (λ0 ◦ t0 ) · m0 e0 c0 , 1C10 ◦ Φ
¶
µ
= λ0 · 1m0 De0 c0 ,1 0 E ◦ (1t0 · Φ)
C1
= λ0 ◦ Φ.
The proof on rho is similar.
The three last propositions lead us to the following theorem.
Journal of Homotopy and Related Structures, vol. 1(1), 2006
75
Theorem 3. Let C be a 2-category, and consider C, C 0 two pseudo-categories in
C. The data:
• objects: pseudo-functors from C to C 0 ;
• morphisms: natural transformations (between pseudo-functors from C to C 0 );
• pseudo-morphisms: pseudo-natural transformations (between pseudo-functors
from C to C 0 );
• cells: pseudo-modifications (between such natural and pseudo-natural transformations);
form a pseudo-category (in Cat).
Proof. Natural transformations and pseudo-functors form a category: theorem 1.
pseudo-modifications and pseudo-natural transformations also form a category: the
composition is associative and has identities (that inherit the structure of 2-cells of
the ambient 2-category).
For every pseudo-natural transformation T = (t, τ ) : F −→ G, the identity
pseudo-modification is 1T = 1t
F
1
²
F
/G
T
®¶
1T
T
²1
/G
.
For each pair of pseudo-composable pseudo-modifications Φ, Ψ, there is a (well
defined - proposition 2) pseudo-composition Φ ⊗ Ψ = m0 hΦ, Ψi satisfying (1.8)
(ΦΦ0 ) ⊗ (ΨΨ0 ) = m0 hΦΦ0 , ΨΨ0 i
(Φ ⊗ Ψ) (Φ0 ⊗ Ψ0 ) = (m0 hΦ, Ψi) (m0 hΦ0 , Ψ0 i)
= (1m0 1m0 ) ◦ (hΦ, Ψi hΦ0 , Ψ0 i)
= m0 hΦΦ0 , ΨΨ0 i ;
and 1T ⊗S = 1T ⊗ 1S ,
1mht,si = 1m ◦ 1ht,si = 1m ◦ h1t , 1s i = m h1t , 1s i .
For each natural transformation θ : F −→ G there is a pseudo-modification
F
θ
²
G
idF
®¶
idθ
idG
/F
²θ
/G
with idθ = e0 θ0 , satisfying
id1F = e0 1F0 = 1e0 F0 = 1idF ,
idθ0 θ = e0 ◦ (θ00 θ0 ) = (e0 ◦ θ00 ) (e0 ◦ θ0 ) = idθ0 idθ .
Journal of Homotopy and Related Structures, vol. 1(1), 2006
76
By Proposition 3 there is a special pseudo-modification αT,U,S = α hT, U, Si for
each triple of composable pseudo-natural transformations T, U, S, natural in each
component and satisfying the pentagon coherence condition.
By Proposition 5 there are two special pseudo-modifications λT , ρT to each
pseudo-natural transformation T : F −→ G, natural in T and satisfying the triangle coherence condition.
5.
Conclusion and final remarks
The mathematical object PsCat that we have just defined has the following
structure:
• objects: A, B, C, ...
• morphisms: f : A −→ B, ...
• 2-cells: θ : f −→ g, ...(f, g : A −→ B)
• pseudo-cells: f
• tetra cells:
T
/ g , ...
/g
f
T
²
f0
®¶ 0
θ
Φ
T
²θ
/ g0
0
, ...
where objects, morphisms and 2-cells form a 2-category and for each pair of
objects A, B, the morphisms, 2-cells, pseudo-cells and tetra cells from A to B form
a pseudo-category.
Two questions arise at this moment:
- What is happening from PsCat(B, C)×PsCat(A, B) to PsCat(A, C)?
- What is the relation between PsCat(A × B, C) and PsCat(A,PsCAT(B, C))?
The answer to the second question is easy to find out. If starting with a pseudofunctor in PsCat(A × B, C), say
h : A × B −→ C,
¡
¢
by going to PsCat A, C B and coming back we will obtain either
h (c, g) ⊗ h (f, b)
or
h (f, d) ⊗ h (a, g)
Journal of Homotopy and Related Structures, vol. 1(1), 2006
77
instead of h (f, g) as displayed in the diagram below
(a, b)
(f,g)
²
(c, d)
h(f,b)
/ h(c, b)
h(a, b)
FF
FF
FF
FF h(f,g)µ
FF
+3 h(c,g)
h(a,g)
FF
FF
FF
µ
FF
²
" ²
®¶
/ h(c, d)
h(a, d)
h(f,d)
And since they are all isomorphic via µ and τ we have that the relation is an
equivalence of categories.
A similar phenomena happens when trying to define horizontal composition of
pseudo-natural transformations (while trying to answer the first question): there
are two equally good ways to define a horizontal composition and they differ by an
isomorphism.
Let C be a 2-category and C, C 0 , C 00 pseudo-categories in C, consider S, T pseudonatural transformations as in
F
F0
G
G
−−−−→ −−−−→
C ↓ T C 0 ↓ S C 00
−−−−→ −−−−
→
0
there are two possibilities to define horizontal composition
S ◦w1 T = m00 hsG0 , F1 t0 i
and
S ◦w2 T = m00 hG01 t, sF0 i
as displayed in the following picture
e
C1 ←−−−− C0




F1 yG1 .t F0 yG0
e0
C10 ←−−−− C00
.




F10 yG01 .s F00 yG00
e00
C100 ←−−−− C000
Hence we have two isomorphic functors from PsCat(B, C)×PsCat(A, B)
to PsCat(A, C) both defining a horizontal composition.
We note that this behaviour, of composition beeing defined up to isomorphism,
also occurs while trying to compose homotopies. So one can expect further relations
between the theory of pseudo-categories and homotopy theory to be investigated.
For instance the category Top itself may be viewed as a structure with objects (spaces), morphisms (continuous mappings), 2-cells (homotopy classes of homotopies), pseudo-cells (simple homotopies) and tetra cells (homotopies between
homotopies).
Journal of Homotopy and Related Structures, vol. 1(1), 2006
78
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number 47, pages 1-77, Springer-Verlag, Berlin, 1967.
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Notes in Mathematics, Springer-Verlag, 1974
[3] T. Leinster, Higher Operads, Higher Categories, London Mathematical Society Lecture Notes Series, Cambridge University Press, 2003 (electronic version).
[4] S. MacLane, Categories for the working Mathematician, Springer-Verlag,
1998, 2nd edition.
[5] N. Martins-Ferreira, Internal Bicategories in Ab, Preprint CM03/I-24, Aveiro
Universitiy, 2003.
[6] N. Martins-Ferreira, Internal Weak Categories in Additive 2-Categories with
Kernels, Fields Institute Communications, Volume 43, p.387-410, 2004.
[7] N. Martins-Ferreira, Weak categories in Grp, Unpublished.
[8] R. Paré and M. Grandis, Adjoints for Double Categories, Cahiers Topologie
et Géométrie Diferéntielle Catégoriques, XLV(3),2004, 193-240.
[9] R. Paré and M. Grandis, Limits in Double Categories, Cahiers Topologie et
Géométrie Diferéntielle Catégoriques, XL(3), 1999, 162-220.
[10] J. Power, 2-Categories, BRICS Notes Series, NS-98-7.
[11] R.H. Street, Cosmoi of internal categories, Trans. Amer. Math. Soc. 258,
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This article may be accessed via WWW at http://jhrs.rmi.acnet.ge
N. Martins-Ferreira
nelsonmf@estg.ipleiria.pt
http://www.estg.ipleiria.pt/~nelsonmf
Polytechnic Institute of Leiria,
Portugal