Terminal coalgebras
Eugenia Cheng and Tom Leinster
University of Sheffield and University of Glasgow
March 2008
1. 2. Some theory of terminal coalgebras 3. Weak n-categories 4. Operads 5. Trimble-like n-categories 6. Trimble-like ω-categories via terminal coalgebras
Plan
1. Introduction to terminal coalgebras
2. 3. Weak n-categories 4. Operads 5. Trimble-like n-categories 6. Trimble-like ω-categories via terminal coalgebras
Plan
1. Introduction to terminal coalgebras 2. Some theory of terminal coalgebras
2. 4. Operads 5. Trimble-like n-categories 6. Trimble-like ω-categories via terminal coalgebras
Plan
1. Introduction to terminal coalgebras 2. Some theory of terminal coalgebras 3. Weak n-categories
2. 5. Trimble-like n-categories 6. Trimble-like ω-categories via terminal coalgebras
Plan
1. Introduction to terminal coalgebras 2. Some theory of terminal coalgebras 3. Weak n-categories 4. Operads
2. 6. Trimble-like ω-categories via terminal coalgebras
Plan
1. Introduction to terminal coalgebras 2. Some theory of terminal coalgebras 3. Weak n-categories 4. Operads 5. Trimble-like n-categories
2. Plan
1. Introduction to terminal coalgebras 2. Some theory of terminal coalgebras 3. Weak n-categories 4. Operads 5. Trimble-like n-categories 6. Trimble-like ω-categories via terminal coalgebras
2. A coalgebra for an endofunctor F : C −→ C consists of • an object A ∈ C A • a morphism FA satisfying no axioms.
1. Introduction to terminal coalgebras
3. A • a morphism FA satisfying no axioms.
1. Introduction to terminal coalgebras
A coalgebra for an endofunctor F : C −→ C consists of • an object A ∈ C
3. satisfying no axioms.
1. Introduction to terminal coalgebras
A coalgebra for an endofunctor F : C −→ C consists of • an object A ∈ C A • a morphism FA
3. 1. Introduction to terminal coalgebras
A coalgebra for an endofunctor F : C −→ C consists of • an object A ∈ C A • a morphism FA satisfying no axioms.
3. h AB/
/ FA F h FB
so we can look for terminal coalgebras.
1. Introduction to terminal coalgebras
Coalgebras for F form a category with the obvious morphisms
4. so we can look for terminal coalgebras.
1. Introduction to terminal coalgebras
Coalgebras for F form a category with the obvious morphisms
h AB/
/ FA F h FB
4. 1. Introduction to terminal coalgebras
Coalgebras for F form a category with the obvious morphisms
h AB/
/ FA F h FB so we can look for terminal coalgebras.
4. 1. Introduction to terminal coalgebras
Example 1 Given a set M we have an endofunctor
Set −→M× Set A 7→ M × A
5. 1. Introduction to terminal coalgebras
Example 1 Given a set M we have an endofunctor
Set −→M× Set A 7→ M × A
A coalgebra for this is a function
5. 1. Introduction to terminal coalgebras
Example 1 Given a set M we have an endofunctor
Set −→M× Set A 7→ M × A
A coalgebra for this is a function
(m,f) A −→ M × A a 7→ (m(a), f(a))
5. 1. Introduction to terminal coalgebras
Example 1 Given a set M we have an endofunctor
Set −→M× Set A 7→ M × A
5. 1. Introduction to terminal coalgebras
Example 1 Given a set M we have an endofunctor
Set −→M× Set A 7→ M × A
The terminal coalgebra is given by the set M N of “infinite words” in M
(m1, m2, m3,...)
5. We need a map M N
M × M N
—we have a canonical isomorphism.
1. Introduction to terminal coalgebras
To see that this is a coalgebra:
6. —we have a canonical isomorphism.
1. Introduction to terminal coalgebras
To see that this is a coalgebra: We need a map M N
M × M N
6. 1. Introduction to terminal coalgebras
To see that this is a coalgebra: We need a map M N
M × M N
—we have a canonical isomorphism.
6. Given any coalgebra we need a unique map t A A / M N
(m,f) / N M × A M × A 1×t M × M and we have t : a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
To see that this is terminal:
7. we need a unique map t A / M N
/ N M × A 1×t M × M and we have t : a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
To see that this is terminal: Given any coalgebra A
(m,f) M × A
7. and we have t : a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
To see that this is terminal: Given any coalgebra we need a unique map t A A / M N
(m,f) / N M × A M × A 1×t M × M
7. 1. Introduction to terminal coalgebras
To see that this is terminal: Given any coalgebra we need a unique map t A A / M N
(m,f) / N M × A M × A 1×t M × M and we have t : a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
7. a m(a) f(a) m(f(a)) f 2(a) m(f 2(a)) f 3(a) m(f 3(a)) f 4(a) . .
a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
screen memory
8. m(a) f(a) m(f(a)) f 2(a) m(f 2(a)) f 3(a) m(f 3(a)) f 4(a) . .
a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
screen memory a
8. m(f(a)) f 2(a) m(f 2(a)) f 3(a) m(f 3(a)) f 4(a) . .
a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
screen memory a m(a) f(a)
8. m(f 2(a)) f 3(a) m(f 3(a)) f 4(a) . .
a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
screen memory a m(a) f(a) m(f(a)) f 2(a)
8. m(f 3(a)) f 4(a) . .
a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
screen memory a m(a) f(a) m(f(a)) f 2(a) m(f 2(a)) f 3(a)
8. . .
a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
screen memory a m(a) f(a) m(f(a)) f 2(a) m(f 2(a)) f 3(a) m(f 3(a)) f 4(a)
8. a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
1. Introduction to terminal coalgebras
screen memory a m(a) f(a) m(f(a)) f 2(a) m(f 2(a)) f 3(a) m(f 3(a)) f 4(a) . .
8. 1. Introduction to terminal coalgebras
screen memory a m(a) f(a) m(f(a)) f 2(a) m(f 2(a)) f 3(a) m(f 3(a)) f 4(a) . . a 7→ m(a), m(f(a)), m(f 2(a)), m(f 3(a)),...
8. A coalgebra for this is a function
f A −→ FA
a 7→ (a1, a2, . . . , an)
The terminal coalgebra is given by the set Tr∞ of infinite trees of finite arity
1. Introduction to terminal coalgebras
Example 2 Let F be the free monoid monad on Set.
9. f A −→ FA
a 7→ (a1, a2, . . . , an)
The terminal coalgebra is given by the set Tr∞ of infinite trees of finite arity
1. Introduction to terminal coalgebras
Example 2 Let F be the free monoid monad on Set. A coalgebra for this is a function
9. a 7→ (a1, a2, . . . , an)
The terminal coalgebra is given by the set Tr∞ of infinite trees of finite arity
1. Introduction to terminal coalgebras
Example 2 Let F be the free monoid monad on Set. A coalgebra for this is a function
f A −→ FA
9. The terminal coalgebra is given by the set Tr∞ of infinite trees of finite arity
1. Introduction to terminal coalgebras
Example 2 Let F be the free monoid monad on Set. A coalgebra for this is a function
f A −→ FA
a 7→ (a1, a2, . . . , an)
9. the set Tr∞ of infinite trees of finite arity
1. Introduction to terminal coalgebras
Example 2 Let F be the free monoid monad on Set. A coalgebra for this is a function
f A −→ FA
a 7→ (a1, a2, . . . , an)
The terminal coalgebra is given by
9. 1. Introduction to terminal coalgebras
Example 2 Let F be the free monoid monad on Set. A coalgebra for this is a function
f A −→ FA
a 7→ (a1, a2, . . . , an)
The terminal coalgebra is given by the set Tr∞ of infinite trees of finite arity
9. We need a map
Tr∞
F (Tr∞) = finite strings of infinite trees
—we have a canonical isomorphism.
1. Introduction to terminal coalgebras
To see that this is a coalgebra:
10. = finite strings of infinite trees
—we have a canonical isomorphism.
1. Introduction to terminal coalgebras
To see that this is a coalgebra: We need a map
Tr∞
F (Tr∞)
10. —we have a canonical isomorphism.
1. Introduction to terminal coalgebras
To see that this is a coalgebra: We need a map
Tr∞
F (Tr∞) = finite strings of infinite trees
10. 1. Introduction to terminal coalgebras
To see that this is a coalgebra: We need a map
Tr∞
F (Tr∞) = finite strings of infinite trees
—we have a canonical isomorphism.
10. Given any coalgebra we need a unique map t A A / Tr∞
f ∞ / F (Tr ) FA FA F t —we take a ∈ A and send it to the tree resulting from “infinite iteration of the programme”
1. Introduction to terminal coalgebras
To see that this is terminal:
11. we need a unique map t A / Tr∞
∞ / F (Tr ) FA F t —we take a ∈ A and send it to the tree resulting from “infinite iteration of the programme”
1. Introduction to terminal coalgebras
To see that this is terminal: Given any coalgebra A
f FA
11. —we take a ∈ A and send it to the tree resulting from “infinite iteration of the programme”
1. Introduction to terminal coalgebras
To see that this is terminal: Given any coalgebra we need a unique map t A A / Tr∞
f ∞ / F (Tr ) FA FA F t
11. 1. Introduction to terminal coalgebras
To see that this is terminal: Given any coalgebra we need a unique map t A A / Tr∞
f ∞ / F (Tr ) FA FA F t —we take a ∈ A and send it to the tree resulting from “infinite iteration of the programme”
11. Lemma (Lambek)
If A is a terminal coalgebra for F f FA then f is an isomorphism.
2. Some theory of terminal coalgebras
12. If A is a terminal coalgebra for F f FA then f is an isomorphism.
2. Some theory of terminal coalgebras
Lemma (Lambek)
12. then f is an isomorphism.
2. Some theory of terminal coalgebras
Lemma (Lambek)
If A is a terminal coalgebra for F f FA
12. 2. Some theory of terminal coalgebras
Lemma (Lambek)
If A is a terminal coalgebra for F f FA then f is an isomorphism.
12. Theorem (Ad´amek)
We can construct the terminal coalgebra as the limit of the following diagram:
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
provided there is a terminal object 1, the limit exists, F preserves it
2. Some theory of terminal coalgebras
13. We can construct the terminal coalgebra as the limit of the following diagram:
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
provided there is a terminal object 1, the limit exists, F preserves it
2. Some theory of terminal coalgebras
Theorem (Ad´amek)
13. F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
provided there is a terminal object 1, the limit exists, F preserves it
2. Some theory of terminal coalgebras
Theorem (Ad´amek)
We can construct the terminal coalgebra as the limit of the following diagram:
13. provided there is a terminal object 1, the limit exists, F preserves it
2. Some theory of terminal coalgebras
Theorem (Ad´amek)
We can construct the terminal coalgebra as the limit of the following diagram:
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
13. 2. Some theory of terminal coalgebras
Theorem (Ad´amek)
We can construct the terminal coalgebra as the limit of the following diagram:
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 provided there is a terminal object 1, the limit exists, F preserves it
13. 2. Some theory of terminal coalgebras
Example 1 revisited Given a set M we considered the endofunctor
Set −→M× Set A 7→ M × A
14. 2. Some theory of terminal coalgebras
Example 1 revisited Given a set M we considered the endofunctor
Set −→M× Set A 7→ M × A and saw that the terminal coalgebra was given by the set M N of “infinite words” in M
(m1, m2, m3,...)
14. 2. Some theory of terminal coalgebras
Example 1 revisited Given a set M we considered the endofunctor
Set −→M× Set A 7→ M × A
We can construct a terminal coalgebra as the limit of
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
14. 2. Some theory of terminal coalgebras
Example 1 revisited Given a set M we considered the endofunctor
Set −→M× Set A 7→ M × A
We can construct a terminal coalgebra as the limit of
M 3×! M 2×! M×! ! ··· / M 3 × 1 / M 2 × 1 / M × 1 / 1
14. 2. Some theory of terminal coalgebras
Example 1 revisited Given a set M we considered the endofunctor
Set −→M× Set A 7→ M × A
We can construct a terminal coalgebra as the limit of
M 3×! M 2×! M×! ! ··· / M 3 / M 2 / M / 1
14. 2. Some theory of terminal coalgebras
Example 1 revisited Given a set M we considered the endofunctor
Set −→M× Set A 7→ M × A
We can construct a terminal coalgebra as the limit of
M 3×! M 2×! M×! ! ··· / M 3 / M 2 / M / 1 which does indeed give infinite words in M.
14. We can construct the terminal coalgebra as the limit of the following diagram:
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 which does indeed give the set of infinite trees of finite arity.
2. Some theory of terminal coalgebras
Example 2 revisited Let F be the free monoid monad on Set.
15. which does indeed give the set of infinite trees of finite arity.
2. Some theory of terminal coalgebras
Example 2 revisited Let F be the free monoid monad on Set.
We can construct the terminal coalgebra as the limit of the following diagram:
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
15. 2. Some theory of terminal coalgebras
Example 2 revisited Let F be the free monoid monad on Set.
We can construct the terminal coalgebra as the limit of the following diagram:
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 which does indeed give the set of infinite trees of finite arity.
15. V 7→ V-Gph
A V-graph A consists of • a set obA • for all x, y ∈ obA an object A(x, y) ∈ V
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat
16. A V-graph A consists of • a set obA • for all x, y ∈ obA an object A(x, y) ∈ V
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
16. • a set obA • for all x, y ∈ obA an object A(x, y) ∈ V
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
A V-graph A consists of
16. • for all x, y ∈ obA an object A(x, y) ∈ V
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
A V-graph A consists of • a set obA
16. 2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
A V-graph A consists of • a set obA • for all x, y ∈ obA an object A(x, y) ∈ V
16. The terminal coalgebra is given by the category of globular sets
s s s s s / / / / / ······/ A(n) / / A(2) / A(1) / A(0) t t t t t
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
17. the category of globular sets
s s s s s / / / / / ······/ A(n) / / A(2) / A(1) / A(0) t t t t t
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
The terminal coalgebra is given by
17. s s s s s / / / / / ······/ A(n) / / A(2) / A(1) / A(0) t t t t t
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
The terminal coalgebra is given by the category of globular sets
17. 2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
The terminal coalgebra is given by the category of globular sets
s s s s s / / / / / ······/ A(n) / / A(2) / A(1) / A(0) t t t t t
17. We write GSet or ω-Gph and note that Lambek’s Lemma holds
ω-Gph ∼= (ω-Gph)-Gph.
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
18. and note that Lambek’s Lemma holds
ω-Gph ∼= (ω-Gph)-Gph.
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
We write GSet or ω-Gph
18. ω-Gph ∼= (ω-Gph)-Gph.
2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
We write GSet or ω-Gph and note that Lambek’s Lemma holds
18. 2. Some theory of terminal coalgebras
Example 3 There is an endofunctor Cat −→ Cat V 7→ V-Gph
We write GSet or ω-Gph and note that Lambek’s Lemma holds
ω-Gph ∼= (ω-Gph)-Gph.
18. • For each n we have a category n-GSet of n-truncated globular sets
2. Some theory of terminal coalgebras
Using Ad´amek’s construction
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets
s s s s s / / / / / A(n) / A(n − 1) / ··· / A(2) / A(1) / A(0) t t t t t
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets • F 1 = 1-Gph ∼= Set
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets • F 1 = 1-Gph ∼= Set • F n1 = (n − 1)-GSet-Gph = n-GSet
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets • F 1 = 1-Gph ∼= Set • F n1 = (n − 1)-GSet-Gph = n-GSet
The limit diagram
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets • F 1 = 1-Gph ∼= Set • F n1 = (n − 1)-GSet-Gph = n-GSet
The limit diagram becomes
! ··· / 2-GSet / 1-GSet / 0-GSet / 1
19. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• For each n we have a category n-GSet of n-truncated globular sets • F 1 = 1-Gph ∼= Set • F n1 = (n − 1)-GSet-Gph = n-GSet
The limit diagram becomes
! ··· / 2-GSet / 1-GSet / 0-GSet / 1 where each morphism here is truncation.
19. The terminal coalgebra is given by the category ω-Cat of strict ω-categories. Again, we note that Lambek’s Lemma holds: ω-Cat ∼= (ω-Cat)-Cat.
2. Some theory of terminal coalgebras
Example 4 (Simpson) There is an endofunctor SymMonCat −→ SymMonCat V 7→ V-Cat
20. the category ω-Cat of strict ω-categories. Again, we note that Lambek’s Lemma holds: ω-Cat ∼= (ω-Cat)-Cat.
2. Some theory of terminal coalgebras
Example 4 (Simpson) There is an endofunctor SymMonCat −→ SymMonCat V 7→ V-Cat
The terminal coalgebra is given by
20. Again, we note that Lambek’s Lemma holds: ω-Cat ∼= (ω-Cat)-Cat.
2. Some theory of terminal coalgebras
Example 4 (Simpson) There is an endofunctor SymMonCat −→ SymMonCat V 7→ V-Cat
The terminal coalgebra is given by the category ω-Cat of strict ω-categories.
20. 2. Some theory of terminal coalgebras
Example 4 (Simpson) There is an endofunctor SymMonCat −→ SymMonCat V 7→ V-Cat
The terminal coalgebra is given by the category ω-Cat of strict ω-categories. Again, we note that Lambek’s Lemma holds: ω-Cat ∼= (ω-Cat)-Cat.
20. • F 1 = 1-Cat ∼= Set • F n1 = (n − 1)-Cat-Cat = n-Cat
2. Some theory of terminal coalgebras
Using Ad´amek’s construction
21. • F n1 = (n − 1)-Cat-Cat = n-Cat
2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• F 1 = 1-Cat ∼= Set
21. The limit diagram
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• F 1 = 1-Cat ∼= Set • F n1 = (n − 1)-Cat-Cat = n-Cat
21. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• F 1 = 1-Cat ∼= Set • F n1 = (n − 1)-Cat-Cat = n-Cat
The limit diagram
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
21. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• F 1 = 1-Cat ∼= Set • F n1 = (n − 1)-Cat-Cat = n-Cat
The limit diagram becomes
! ··· / 2-Cat / 1-Cat / 0-Cat / 1
21. 2. Some theory of terminal coalgebras
Using Ad´amek’s construction
• F 1 = 1-Cat ∼= Set • F n1 = (n − 1)-Cat-Cat = n-Cat
The limit diagram becomes
! ··· / 2-Cat / 1-Cat / 0-Cat / 1 where each morphism here is truncation.
21. Aim —to apply this to Trimble’s version of weak n-categories.
2. Some theory of terminal coalgebras
Idea This gives us a way of constructing infinite versions of gadgets whose finite versions we can construct simply by induction.
22. 2. Some theory of terminal coalgebras
Idea This gives us a way of constructing infinite versions of gadgets whose finite versions we can construct simply by induction.
Aim —to apply this to Trimble’s version of weak n-categories.
22. For strict n-categories we can just enrich in (n − 1)-Cat
n-Cat := ((n − 1)-Cat)-Cat.
For Trimble’s weak n-categories we • enrich in (n − 1)-Cat, and • weaken the composition using an operad
What does “weaken” mean?
3. Weak n-categories
23. n-Cat := ((n − 1)-Cat)-Cat.
For Trimble’s weak n-categories we • enrich in (n − 1)-Cat, and • weaken the composition using an operad
What does “weaken” mean?
3. Weak n-categories
For strict n-categories we can just enrich in (n − 1)-Cat
23. For Trimble’s weak n-categories we • enrich in (n − 1)-Cat, and • weaken the composition using an operad
What does “weaken” mean?
3. Weak n-categories
For strict n-categories we can just enrich in (n − 1)-Cat
n-Cat := ((n − 1)-Cat)-Cat.
23. • enrich in (n − 1)-Cat, and • weaken the composition using an operad
What does “weaken” mean?
3. Weak n-categories
For strict n-categories we can just enrich in (n − 1)-Cat
n-Cat := ((n − 1)-Cat)-Cat.
For Trimble’s weak n-categories we
23. and • weaken the composition using an operad
What does “weaken” mean?
3. Weak n-categories
For strict n-categories we can just enrich in (n − 1)-Cat
n-Cat := ((n − 1)-Cat)-Cat.
For Trimble’s weak n-categories we • enrich in (n − 1)-Cat,
23. • weaken the composition using an operad
What does “weaken” mean?
3. Weak n-categories
For strict n-categories we can just enrich in (n − 1)-Cat
n-Cat := ((n − 1)-Cat)-Cat.
For Trimble’s weak n-categories we • enrich in (n − 1)-Cat, and
23. What does “weaken” mean?
3. Weak n-categories
For strict n-categories we can just enrich in (n − 1)-Cat
n-Cat := ((n − 1)-Cat)-Cat.
For Trimble’s weak n-categories we • enrich in (n − 1)-Cat, and • weaken the composition using an operad
23. 3. Weak n-categories
For strict n-categories we can just enrich in (n − 1)-Cat
n-Cat := ((n − 1)-Cat)-Cat.
For Trimble’s weak n-categories we • enrich in (n − 1)-Cat, and • weaken the composition using an operad
What does “weaken” mean?
23. • 0-cells ·
• 1-cells ··/
• 2-cells ·· A There are various kinds of composition:
···/ / .. / ... G D D
3. Weak n-categories
A bicategory has
24. • 1-cells ··/
• 2-cells ·· A There are various kinds of composition:
···/ / .. / ... G D D
3. Weak n-categories
A bicategory has • 0-cells ·
24. • 2-cells ·· A There are various kinds of composition:
···/ / .. / ... G D D
3. Weak n-categories
A bicategory has • 0-cells ·
• 1-cells ··/
24. There are various kinds of composition:
···/ / .. / ... G D D
3. Weak n-categories
A bicategory has • 0-cells ·
• 1-cells ··/
• 2-cells ·· A
24. ···/ / .. / ... G D D
3. Weak n-categories
A bicategory has • 0-cells ·
• 1-cells ··/
• 2-cells ·· A There are various kinds of composition:
24. .. / ... G D D
3. Weak n-categories
A bicategory has • 0-cells ·
• 1-cells ··/
• 2-cells ·· A There are various kinds of composition:
···/ /
24. ... D D
3. Weak n-categories
A bicategory has • 0-cells ·
• 1-cells ··/
• 2-cells ·· A There are various kinds of composition:
.. ···/ / / G
24. 3. Weak n-categories
A bicategory has • 0-cells ·
• 1-cells ··/
• 2-cells ·· A There are various kinds of composition:
···/ / .. / ... G D D
24. Unlike in a strict 2-category we do not have
(hg)f = h(gf).
That is, given a composable diagram
f g h a / b / c / d
we have two composites.
3. Weak n-categories
Axioms in a bicategory
25. That is, given a composable diagram
f g h a / b / c / d
we have two composites.
3. Weak n-categories
Axioms in a bicategory Unlike in a strict 2-category we do not have
(hg)f = h(gf).
25. we have two composites.
3. Weak n-categories
Axioms in a bicategory Unlike in a strict 2-category we do not have
(hg)f = h(gf).
That is, given a composable diagram
f g h a / b / c / d
25. 3. Weak n-categories
Axioms in a bicategory Unlike in a strict 2-category we do not have
(hg)f = h(gf).
That is, given a composable diagram
f g h a / b / c / d we have two composites.
25. we have many composites.
Given a diagram
·· / ····/ ·· D D B JL we have very many composites.
3. Weak n-categories Given a diagram
f1 f2 fk a0 / a1 / ... / ak−1 / ak
26. many composites.
Given a diagram
·· / ····/ ·· D D B JL we have very many composites.
3. Weak n-categories Given a diagram
f1 f2 fk a0 / a1 / ... / ak−1 / ak we have
26. Given a diagram
·· / ····/ ·· D D B JL we have very many composites.
3. Weak n-categories Given a diagram
f1 f2 fk a0 / a1 / ... / ak−1 / ak we have many composites.
26. we have very many composites.
3. Weak n-categories Given a diagram
f1 f2 fk a0 / a1 / ... / ak−1 / ak we have many composites.
Given a diagram
·· / ····/ ·· D D B JL
26. very many composites.
3. Weak n-categories Given a diagram
f1 f2 fk a0 / a1 / ... / ak−1 / ak we have many composites.
Given a diagram
·· / ····/ ·· D D B JL we have
26. 3. Weak n-categories Given a diagram
f1 f2 fk a0 / a1 / ... / ak−1 / ak we have many composites.
Given a diagram
·· / ····/ ·· D D B JL we have very many composites.
26. 3. Weak n-categories
Idea We will keep track of all these composites using operads.
27. Let V be a symmetric monoidal category. An operad P in V is given by • for each integer k ≥ 0 an object P (k) ∈ V • composition morphisms
P (k)⊗P (m1)⊗· · ·⊗P (mk) −→ P (m1+···+mk)
satisfying unit and associativity axioms.
4. Operads
28. An operad P in V is given by • for each integer k ≥ 0 an object P (k) ∈ V • composition morphisms
P (k)⊗P (m1)⊗· · ·⊗P (mk) −→ P (m1+···+mk)
satisfying unit and associativity axioms.
4. Operads
Let V be a symmetric monoidal category.
28. • for each integer k ≥ 0 an object P (k) ∈ V • composition morphisms
P (k)⊗P (m1)⊗· · ·⊗P (mk) −→ P (m1+···+mk)
satisfying unit and associativity axioms.
4. Operads
Let V be a symmetric monoidal category. An operad P in V is given by
28. • composition morphisms
P (k)⊗P (m1)⊗· · ·⊗P (mk) −→ P (m1+···+mk)
satisfying unit and associativity axioms.
4. Operads
Let V be a symmetric monoidal category. An operad P in V is given by • for each integer k ≥ 0 an object P (k) ∈ V
28. satisfying unit and associativity axioms.
4. Operads
Let V be a symmetric monoidal category. An operad P in V is given by • for each integer k ≥ 0 an object P (k) ∈ V • composition morphisms
P (k)⊗P (m1)⊗· · ·⊗P (mk) −→ P (m1+···+mk)
28. 4. Operads
Let V be a symmetric monoidal category. An operad P in V is given by • for each integer k ≥ 0 an object P (k) ∈ V • composition morphisms
P (k)⊗P (m1)⊗· · ·⊗P (mk) −→ P (m1+···+mk) satisfying unit and associativity axioms.
28. 4. Operads
In pictures:
29. 4. Operads
In pictures: An operation in P (3) can be pictured as
29. 4. Operads
In pictures: An operation in P (3) can be pictured as
EE y EE yy Eyy
29. 4. Operads
In pictures: Operad composition then looks like
29. 4. Operads
In pictures: Operad composition then looks like
99 Õ EE y 99 Õ 44 99 ÕÕ EE yy 99 ÕÕ 4 Õ Eyy Õ 44 44 AA } 4 A } 44 AA }} / 4 AA }} 44 AA }} 4 A } 44 AA }} 4 A}} 4
29. 4. Operads
Typical examples of V are • Top • sSet • Cat In all our examples, ⊗ will be ×.
30. An algebra for an operad P in V is given by • an underlying object A ∈ V • for all k ≥ 0 an action
P (k) × Ak −→ A
interacting well with operad composition.
4. Operads
Algebras for operads
31. • an underlying object A ∈ V • for all k ≥ 0 an action
P (k) × Ak −→ A
interacting well with operad composition.
4. Operads
Algebras for operads An algebra for an operad P in V is given by
31. • for all k ≥ 0 an action
P (k) × Ak −→ A
interacting well with operad composition.
4. Operads
Algebras for operads An algebra for an operad P in V is given by • an underlying object A ∈ V
31. interacting well with operad composition.
4. Operads
Algebras for operads An algebra for an operad P in V is given by • an underlying object A ∈ V • for all k ≥ 0 an action
P (k) × Ak −→ A
31. 4. Operads
Algebras for operads An algebra for an operad P in V is given by • an underlying object A ∈ V • for all k ≥ 0 an action
P (k) × Ak −→ A interacting well with operad composition.
31. Idea
A(V,P )-category will be a cross between
• a V-category, and • a P -algebra. The underlying data is a V-graph but composition is like a P -algebra action.
5. Trimble-like weak n-categories
32. A(V,P )-category will be a cross between
• a V-category, and • a P -algebra. The underlying data is a V-graph but composition is like a P -algebra action.
5. Trimble-like weak n-categories
Idea
32. • a V-category, and • a P -algebra. The underlying data is a V-graph but composition is like a P -algebra action.
5. Trimble-like weak n-categories
Idea
A(V,P )-category will be a cross between
32. • a P -algebra. The underlying data is a V-graph but composition is like a P -algebra action.
5. Trimble-like weak n-categories
Idea
A(V,P )-category will be a cross between
• a V-category, and
32. The underlying data is a V-graph but composition is like a P -algebra action.
5. Trimble-like weak n-categories
Idea
A(V,P )-category will be a cross between
• a V-category, and • a P -algebra.
32. but composition is like a P -algebra action.
5. Trimble-like weak n-categories
Idea
A(V,P )-category will be a cross between
• a V-category, and • a P -algebra. The underlying data is a V-graph
32. 5. Trimble-like weak n-categories
Idea
A(V,P )-category will be a cross between
• a V-category, and • a P -algebra. The underlying data is a V-graph but composition is like a P -algebra action.
32. A(ak−1, ak) × · · · × A(a0, a1) −→ A(a0, ak)
• P -algebra action: P (k) × A × · · · × A −→ A
• Composition in a (V,P )-category:
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
5. Trimble-like weak n-categories
• Composition in an ordinary V-category:
33. • P -algebra action: P (k) × A × · · · × A −→ A
• Composition in a (V,P )-category:
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
5. Trimble-like weak n-categories
• Composition in an ordinary V-category:
A(ak−1, ak) × · · · × A(a0, a1) −→ A(a0, ak)
33. P (k) × A × · · · × A −→ A
• Composition in a (V,P )-category:
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
5. Trimble-like weak n-categories
• Composition in an ordinary V-category:
A(ak−1, ak) × · · · × A(a0, a1) −→ A(a0, ak)
• P -algebra action:
33. • Composition in a (V,P )-category:
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
5. Trimble-like weak n-categories
• Composition in an ordinary V-category:
A(ak−1, ak) × · · · × A(a0, a1) −→ A(a0, ak)
• P -algebra action: P (k) × A × · · · × A −→ A
33. P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
5. Trimble-like weak n-categories
• Composition in an ordinary V-category:
A(ak−1, ak) × · · · × A(a0, a1) −→ A(a0, ak)
• P -algebra action: P (k) × A × · · · × A −→ A
• Composition in a (V,P )-category:
33. 5. Trimble-like weak n-categories
• Composition in an ordinary V-category:
A(ak−1, ak) × · · · × A(a0, a1) −→ A(a0, ak)
• P -algebra action: P (k) × A × · · · × A −→ A
• Composition in a (V,P )-category:
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
33. • an underlying V-graph A • composition maps
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
interacting well with the operad structure of P .
5. Trimble-like weak n-categories
Definition
A(V,P )-category consists of
34. • composition maps
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
interacting well with the operad structure of P .
5. Trimble-like weak n-categories
Definition
A(V,P )-category consists of
• an underlying V-graph A
34. interacting well with the operad structure of P .
5. Trimble-like weak n-categories
Definition
A(V,P )-category consists of
• an underlying V-graph A • composition maps
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak)
34. 5. Trimble-like weak n-categories
Definition
A(V,P )-category consists of
• an underlying V-graph A • composition maps
P (k)×A(ak−1, ak)×· · ·×A(a0, a1) −→ A(a0, ak) interacting well with the operad structure of P .
34. • pick a suitable operad P0 ∈ 0-Cat • put 1-Cat = (0-Cat,P0)-Cat • pick a suitable operad P1 ∈ 1-Cat
• put 2-Cat = (1-Cat,P1)-Cat . • .
But what operads Pn are we going to use?
5. Trimble-like weak n-categories
We can then build weak n-categories like this: • put 0-Cat = Set
35. • put 1-Cat = (0-Cat,P0)-Cat • pick a suitable operad P1 ∈ 1-Cat
• put 2-Cat = (1-Cat,P1)-Cat . • .
But what operads Pn are we going to use?
5. Trimble-like weak n-categories
We can then build weak n-categories like this: • put 0-Cat = Set
• pick a suitable operad P0 ∈ 0-Cat
35. • pick a suitable operad P1 ∈ 1-Cat
• put 2-Cat = (1-Cat,P1)-Cat . • .
But what operads Pn are we going to use?
5. Trimble-like weak n-categories
We can then build weak n-categories like this: • put 0-Cat = Set
• pick a suitable operad P0 ∈ 0-Cat • put 1-Cat = (0-Cat,P0)-Cat
35. • put 2-Cat = (1-Cat,P1)-Cat . • .
But what operads Pn are we going to use?
5. Trimble-like weak n-categories
We can then build weak n-categories like this: • put 0-Cat = Set
• pick a suitable operad P0 ∈ 0-Cat • put 1-Cat = (0-Cat,P0)-Cat • pick a suitable operad P1 ∈ 1-Cat
35. . • .
But what operads Pn are we going to use?
5. Trimble-like weak n-categories
We can then build weak n-categories like this: • put 0-Cat = Set
• pick a suitable operad P0 ∈ 0-Cat • put 1-Cat = (0-Cat,P0)-Cat • pick a suitable operad P1 ∈ 1-Cat
• put 2-Cat = (1-Cat,P1)-Cat
35. But what operads Pn are we going to use?
5. Trimble-like weak n-categories
We can then build weak n-categories like this: • put 0-Cat = Set
• pick a suitable operad P0 ∈ 0-Cat • put 1-Cat = (0-Cat,P0)-Cat • pick a suitable operad P1 ∈ 1-Cat
• put 2-Cat = (1-Cat,P1)-Cat . • .
35. 5. Trimble-like weak n-categories
We can then build weak n-categories like this: • put 0-Cat = Set
• pick a suitable operad P0 ∈ 0-Cat • put 1-Cat = (0-Cat,P0)-Cat • pick a suitable operad P1 ∈ 1-Cat
• put 2-Cat = (1-Cat,P1)-Cat . • .
But what operads Pn are we going to use?
35. • start with just one operad E ∈ Top
• take each Pn(k) to be the fundamental n-groupoid of E(k)
So instead of picking one operad Pn for each n, we just have to construct for each n
Πn : Top −→ n-Cat
and this turns out to be easy by induction.
5. Trimble-like weak n-categories
Trimble’s method
36. • take each Pn(k) to be the fundamental n-groupoid of E(k)
So instead of picking one operad Pn for each n, we just have to construct for each n
Πn : Top −→ n-Cat
and this turns out to be easy by induction.
5. Trimble-like weak n-categories
Trimble’s method
• start with just one operad E ∈ Top
36. So instead of picking one operad Pn for each n, we just have to construct for each n
Πn : Top −→ n-Cat
and this turns out to be easy by induction.
5. Trimble-like weak n-categories
Trimble’s method
• start with just one operad E ∈ Top
• take each Pn(k) to be the fundamental n-groupoid of E(k)
36. and this turns out to be easy by induction.
5. Trimble-like weak n-categories
Trimble’s method
• start with just one operad E ∈ Top
• take each Pn(k) to be the fundamental n-groupoid of E(k)
So instead of picking one operad Pn for each n, we just have to construct for each n
Πn : Top −→ n-Cat
36. 5. Trimble-like weak n-categories
Trimble’s method
• start with just one operad E ∈ Top
• take each Pn(k) to be the fundamental n-groupoid of E(k)
So instead of picking one operad Pn for each n, we just have to construct for each n
Πn : Top −→ n-Cat and this turns out to be easy by induction.
36. Each E(k) is the space of continuous endpoint-preserving maps [0, 1] −→ [0, k].
Crucial properties of E: • each E(k) is contractible • E has a natural action on path spaces
E(k) × X(xk−1, xk) × · · · × X(x0, x1) −→ X(x0, xk)
5. Trimble-like weak n-categories
Trimble’s operad E
37. Crucial properties of E: • each E(k) is contractible • E has a natural action on path spaces
E(k) × X(xk−1, xk) × · · · × X(x0, x1) −→ X(x0, xk)
5. Trimble-like weak n-categories
Trimble’s operad E Each E(k) is the space of continuous endpoint-preserving maps [0, 1] −→ [0, k].
37. • each E(k) is contractible • E has a natural action on path spaces
E(k) × X(xk−1, xk) × · · · × X(x0, x1) −→ X(x0, xk)
5. Trimble-like weak n-categories
Trimble’s operad E Each E(k) is the space of continuous endpoint-preserving maps [0, 1] −→ [0, k].
Crucial properties of E:
37. • E has a natural action on path spaces
E(k) × X(xk−1, xk) × · · · × X(x0, x1) −→ X(x0, xk)
5. Trimble-like weak n-categories
Trimble’s operad E Each E(k) is the space of continuous endpoint-preserving maps [0, 1] −→ [0, k].
Crucial properties of E: • each E(k) is contractible
37. 5. Trimble-like weak n-categories
Trimble’s operad E Each E(k) is the space of continuous endpoint-preserving maps [0, 1] −→ [0, k].
Crucial properties of E: • each E(k) is contractible • E has a natural action on path spaces
E(k) × X(xk−1, xk) × · · · × X(x0, x1) −→ X(x0, xk)
37. Let X be a space.
We define an n-category ΠnX as follows: • its objects are just the points of X • (ΠnX)(x, y) := Πn−1 X(x, y) • composition follows from the action of E on path spaces
5. Trimble-like weak n-categories
The fundamental n-groupoid functor
38. We define an n-category ΠnX as follows: • its objects are just the points of X • (ΠnX)(x, y) := Πn−1 X(x, y) • composition follows from the action of E on path spaces
5. Trimble-like weak n-categories
The fundamental n-groupoid functor
Let X be a space.
38. • its objects are just the points of X • (ΠnX)(x, y) := Πn−1 X(x, y) • composition follows from the action of E on path spaces
5. Trimble-like weak n-categories
The fundamental n-groupoid functor
Let X be a space.
We define an n-category ΠnX as follows:
38. • (ΠnX)(x, y) := Πn−1 X(x, y) • composition follows from the action of E on path spaces
5. Trimble-like weak n-categories
The fundamental n-groupoid functor
Let X be a space.
We define an n-category ΠnX as follows: • its objects are just the points of X
38. Πn−1 X(x, y) • composition follows from the action of E on path spaces
5. Trimble-like weak n-categories
The fundamental n-groupoid functor
Let X be a space.
We define an n-category ΠnX as follows: • its objects are just the points of X
• (ΠnX)(x, y) :=
38. • composition follows from the action of E on path spaces
5. Trimble-like weak n-categories
The fundamental n-groupoid functor
Let X be a space.
We define an n-category ΠnX as follows: • its objects are just the points of X • (ΠnX)(x, y) := Πn−1 X(x, y)
38. 5. Trimble-like weak n-categories
The fundamental n-groupoid functor
Let X be a space.
We define an n-category ΠnX as follows: • its objects are just the points of X • (ΠnX)(x, y) := Πn−1 X(x, y) • composition follows from the action of E on path spaces
38. Given a finite product preserving functor Π: Top −→ V
we induce a functor Π+ : Top −→ V-Cat
“do Π locally on the hom objects”
5. Trimble-like weak n-categories Induction for Π in general
39. we induce a functor Π+ : Top −→ V-Cat
“do Π locally on the hom objects”
5. Trimble-like weak n-categories Induction for Π in general
Given a finite product preserving functor Π: Top −→ V
39. “do Π locally on the hom objects”
5. Trimble-like weak n-categories Induction for Π in general
Given a finite product preserving functor Π: Top −→ V we induce a functor Π+ : Top −→ V-Cat
39. 5. Trimble-like weak n-categories Induction for Π in general
Given a finite product preserving functor Π: Top −→ V we induce a functor Π+ : Top −→ V-Cat
“do Π locally on the hom objects”
39. • 0-Cat = Set
Π0 : Top −→ Set X 7→ the set of connected components of X • n-Cat = (n − 1)-Cat, Πn−1E -Cat + Πn = Πn−1
5. Trimble-like weak n-categories
Trimble n-categories by induction
40. Π0 : Top −→ Set X 7→ the set of connected components of X • n-Cat = (n − 1)-Cat, Πn−1E -Cat + Πn = Πn−1
5. Trimble-like weak n-categories
Trimble n-categories by induction
• 0-Cat = Set
40. X 7→ the set of connected components of X • n-Cat = (n − 1)-Cat, Πn−1E -Cat + Πn = Πn−1
5. Trimble-like weak n-categories
Trimble n-categories by induction
• 0-Cat = Set
Π0 : Top −→ Set
40. • n-Cat = (n − 1)-Cat, Πn−1E -Cat + Πn = Πn−1
5. Trimble-like weak n-categories
Trimble n-categories by induction
• 0-Cat = Set
Π0 : Top −→ Set X 7→ the set of connected components of X
40. + Πn = Πn−1
5. Trimble-like weak n-categories
Trimble n-categories by induction
• 0-Cat = Set
Π0 : Top −→ Set X 7→ the set of connected components of X • n-Cat = (n − 1)-Cat, Πn−1E -Cat
40. 5. Trimble-like weak n-categories
Trimble n-categories by induction
• 0-Cat = Set
Π0 : Top −→ Set X 7→ the set of connected components of X • n-Cat = (n − 1)-Cat, Πn−1E -Cat + Πn = Πn−1
40. Idea
• a strict ω-category is a globular set such that each n-truncation is a strict n-category • however if we truncate a weak ω-category we do not get a weak n-category —we get something incoherent at dimension n
So we need to build weak ω-categories from “incoherent n-categories”
6. Trimble-like weak ω-categories
41. • a strict ω-category is a globular set such that each n-truncation is a strict n-category • however if we truncate a weak ω-category we do not get a weak n-category —we get something incoherent at dimension n
So we need to build weak ω-categories from “incoherent n-categories”
6. Trimble-like weak ω-categories
Idea
41. • however if we truncate a weak ω-category we do not get a weak n-category —we get something incoherent at dimension n
So we need to build weak ω-categories from “incoherent n-categories”
6. Trimble-like weak ω-categories
Idea
• a strict ω-category is a globular set such that each n-truncation is a strict n-category
41. —we get something incoherent at dimension n
So we need to build weak ω-categories from “incoherent n-categories”
6. Trimble-like weak ω-categories
Idea
• a strict ω-category is a globular set such that each n-truncation is a strict n-category • however if we truncate a weak ω-category we do not get a weak n-category
41. So we need to build weak ω-categories from “incoherent n-categories”
6. Trimble-like weak ω-categories
Idea
• a strict ω-category is a globular set such that each n-truncation is a strict n-category • however if we truncate a weak ω-category we do not get a weak n-category —we get something incoherent at dimension n
41. 6. Trimble-like weak ω-categories
Idea
• a strict ω-category is a globular set such that each n-truncation is a strict n-category • however if we truncate a weak ω-category we do not get a weak n-category —we get something incoherent at dimension n
So we need to build weak ω-categories from “incoherent n-categories”
41. • 0-iCat = Set
Φ0 : Top −→ Set X 7→ the set of points of X • n-iCat = (n − 1)-iCat, Φn−1E -Cat + Φn = Φn−1
6. Trimble-like weak ω-categories
Incoherent n-categories by induction
42. Φ0 : Top −→ Set X 7→ the set of points of X • n-iCat = (n − 1)-iCat, Φn−1E -Cat + Φn = Φn−1
6. Trimble-like weak ω-categories
Incoherent n-categories by induction
• 0-iCat = Set
42. X 7→ the set of points of X • n-iCat = (n − 1)-iCat, Φn−1E -Cat + Φn = Φn−1
6. Trimble-like weak ω-categories
Incoherent n-categories by induction
• 0-iCat = Set
Φ0 : Top −→ Set
42. • n-iCat = (n − 1)-iCat, Φn−1E -Cat + Φn = Φn−1
6. Trimble-like weak ω-categories
Incoherent n-categories by induction
• 0-iCat = Set
Φ0 : Top −→ Set X 7→ the set of points of X
42. + Φn = Φn−1
6. Trimble-like weak ω-categories
Incoherent n-categories by induction
• 0-iCat = Set
Φ0 : Top −→ Set X 7→ the set of points of X • n-iCat = (n − 1)-iCat, Φn−1E -Cat
42. 6. Trimble-like weak ω-categories
Incoherent n-categories by induction
• 0-iCat = Set
Φ0 : Top −→ Set X 7→ the set of points of X • n-iCat = (n − 1)-iCat, Φn−1E -Cat + Φn = Φn−1
42. where each morphism is truncation.
Question: can we get this as
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 ?
6. Trimble-like weak ω-categories
So we expect to take the following limit ! ··· / 2-iCat / 1-iCat / 0-iCat / 1
43. Question: can we get this as
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 ?
6. Trimble-like weak ω-categories
So we expect to take the following limit ! ··· / 2-iCat / 1-iCat / 0-iCat / 1 where each morphism is truncation.
43. F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 ?
6. Trimble-like weak ω-categories
So we expect to take the following limit ! ··· / 2-iCat / 1-iCat / 0-iCat / 1 where each morphism is truncation.
Question: can we get this as
43. ?
6. Trimble-like weak ω-categories
So we expect to take the following limit ! ··· / 2-iCat / 1-iCat / 0-iCat / 1 where each morphism is truncation.
Question: can we get this as
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
43. 6. Trimble-like weak ω-categories
So we expect to take the following limit ! ··· / 2-iCat / 1-iCat / 0-iCat / 1 where each morphism is truncation.
Question: can we get this as
F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 ?
43. • V is a category with finite products
• Π is a functor Top −→ V preserving finite products. Morphisms are the obvious commuting triangles.
We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
6. Trimble-like weak ω-categories
We work in a category E whose objects are pairs (V, Π) where
44. • Π is a functor Top −→ V preserving finite products. Morphisms are the obvious commuting triangles.
We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
6. Trimble-like weak ω-categories
We work in a category E whose objects are pairs (V, Π) where • V is a category with finite products
44. Morphisms are the obvious commuting triangles.
We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
6. Trimble-like weak ω-categories
We work in a category E whose objects are pairs (V, Π) where • V is a category with finite products
• Π is a functor Top −→ V preserving finite products.
44. We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
6. Trimble-like weak ω-categories
We work in a category E whose objects are pairs (V, Π) where • V is a category with finite products
• Π is a functor Top −→ V preserving finite products. Morphisms are the obvious commuting triangles.
44. (V, Π) 7→ (V, ΠE)-Cat, Π+
6. Trimble-like weak ω-categories
We work in a category E whose objects are pairs (V, Π) where • V is a category with finite products
• Π is a functor Top −→ V preserving finite products. Morphisms are the obvious commuting triangles.
We consider the endofunctor F : E −→ E
44. 6. Trimble-like weak ω-categories
We work in a category E whose objects are pairs (V, Π) where • V is a category with finite products
• Π is a functor Top −→ V preserving finite products. Morphisms are the obvious commuting triangles.
We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
44. For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
45. 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1
45. n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→
45. and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn
45. (1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
45. 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !)
45. (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→
45. 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0)
45. (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→
45. 7→ (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1)
45. (2-iCat, Φ2) 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→
45. 7→ ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2)
45. ···
6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→
45. 6. Trimble-like weak ω-categories We consider the endofunctor F : E −→ E (V, Π) 7→ (V, ΠE)-Cat, Π+
For example F : (n − 1)-Cat, Πn−1 7→ n-Cat, Πn and
(1, !) 7→ (Set, Φ0) 7→ (1-iCat, Φ1) 7→ (2-iCat, Φ2) 7→ ···
45. F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
becomes ! ··· / 2-iCat / 1-iCat / 0-iCat / 1
The terminal coalgebra is indeed the limit we were looking for.
6. Trimble-like weak ω-categories
So the limit
46. becomes ! ··· / 2-iCat / 1-iCat / 0-iCat / 1
The terminal coalgebra is indeed the limit we were looking for.
6. Trimble-like weak ω-categories
So the limit F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1
46. ! ··· / 2-iCat / 1-iCat / 0-iCat / 1
The terminal coalgebra is indeed the limit we were looking for.
6. Trimble-like weak ω-categories
So the limit F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 becomes
46. The terminal coalgebra is indeed the limit we were looking for.
6. Trimble-like weak ω-categories
So the limit F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 becomes ! ··· / 2-iCat / 1-iCat / 0-iCat / 1
46. 6. Trimble-like weak ω-categories
So the limit F 3! F 2! F ! ! ··· / F 31 / F 21 / F 1 / 1 becomes ! ··· / 2-iCat / 1-iCat / 0-iCat / 1
The terminal coalgebra is indeed the limit we were looking for.
46.