A folk model structure on omega-cat

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A folk model structure on omega-cat

joint work with

François Métayer & Krzysztof Worytkiewicz Yves Lafont

Aix-Marseille 2 & CNRS

Institut de Mathématiques de Luminy

HOCAT 2008 CRM, Barcelona

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History

Homology of rewriting (Anick/Squier 1987) Finite derivation type (Squier 1994)

Computads/polygraphs (Street & Power/Burroni 1991) Polygraphic resolutions (Métayer 2003)

Model structure on Cat (Joyal & Tierney 1990) Model structure on 2-Cat (Lack 2002)

2

Strict omega-categories

0-cells 1-cells 2-cells 3-cells ...

Compositions

...

Laws of associativity, units, and interchange

... ...

3

Generating cofibrations

C D p

Acyclic fibrations

i0 ...

O⁰

!O⁰

i1

O¹

!O¹

i2

O²

!O²

i3

O³

!O³

in

!Oⁿ

Oⁿ

x y

such that p u = v.

C

p x D

p yv

For any v : p x ! p y in D,

u

there is u : x ! y in C I:

I-fib:

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Cofibrations Factorization

Theorem (Métayer) Cofibrant objects

are polygraphs.

O⁰

!O¹

o0

O¹

!O²

o1

B A k

C D

p I-fib

B A

f C

p I-fib k I-cof

Any

^"-

functor factorizes as follows:

k0

p0

k1

p1

Crucial example:

...

B

C D p I-cof:

5

Our main result

Theorem

(Lafont, Métayer & Worytkiewicz):

There is a model structure on "-Cat such that:

#

I is a set of generating cofibrations for this structure;

#

the class W of weak equivalences is minimal.

The following model structures are derivable from this:

#

on Cat (Joyal & Tierney 1990);

#

on 2-Cat (Lack 2002);

#

on n-Cat.

They are obtained by Quillen adjunctions (Beke 2001)

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Our strategy

Define the class W of weak equivalences.

Prove the following properties (Smith):

#

I-fib W;

#

W has the 3 for 2 property;

#

I-cof W is closed under pushout, retract, and tranfinite composition;

#

I and W satisfy the solution set condition.

Tools: "-equivalence & reversibility, connections, gluing factorization, immersions, generic squares.

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" -equivalence & reversibility

y

x u

Lemma

If u is reversible, it satisfies a (left) division property:

u v

w such that u ! r ~ s.

(General case: u is a 1-cell and s is a n+1-cell)

u v

w

u s

For all s : u ! v ! u ! w,

r there is r : v ! w

Definition

(by coinduction)

#

x ~ y iff there exists a reversible cell u : x ! y;

#

u : x ! y is reversible iff there exists u : y ! x such that u ! u $ ~ idx and u $ ! u ~ idy.

$

~

u ! u$ ~ ^$^u^!^u

$u

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Weak equivalences

Proposition

The class W of weak equivalences satisfies 3 for 2.

Definition

By a relaxed version of the right lifting property:

x y

such that f u ~ v C

D

A B C A B C A B C

(Some machinery is needed for the third case)

f x f yv

For any v : f x ! f y in D

u

there is u : x ! y in C (not =) W:

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Cylinders & connections

0-cylinder

x

y

1-cylinder

x

y

2-cylinder

x

y

...

Definition

A connection U : x ☎ y is a reversible cylinder.

Lemma

There is a topdown transport for parallel connections:

x

x’

U!

y

y’

!V

z For all z : x ! y,

!W

and a connection W : U ! V | z ☎ z’.

z’

there is z’ : x’ ! y’

z’ is weakly unique.

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Operations on connections

concatenation

Theorem:

Connections in C form an "-category %(C).

x

y

z

trivial cylinder

x

0-composition

x

z

y

t

x

z

1-composition

y

t

...

Concatenation is needed to define n-compositions.

We get the following commutative diagram:

C&

%(C) C "1 "2 C

id id

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Gluing factorization

C D f

Definition

The gluing factorization f = f o f is defined by a pullback:ˆ ˜

%(D) "1

f’

'(f) "1’

Lemma

f W if and only if f I-fib.ˆ

C '(f) D

f fˆ

f˜ where f = "ˆ 2 o f’

Explicit descriptions: ^f(x)

! f(x)

f(x):˜

&(f(x)) f(x)

y

!

'(f):

U

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Immersions

C D C

id

f g D

%(D)h D "1 "2 D fC

g

id D

%(D)h D &

fC f

Definition

f : C ! D is in Z if there are g : D ! C and h : D ! %(D) such that the following three diagrams commute:

Lemma

I-cof W Z W.

Lemma

Z is closed under pushout.

Corollary

I-cof W is closed under pushout.

fC id D

'(f)

D fˆ

f˜

In other words:

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Generic squares

Definition

The generic squares are the following diagrams:

i0

O⁰

!O⁰ i0 O⁰

j0

j0’ P⁰

i1

O¹

!O¹ i1 O¹

j1

j1’ P¹

...

Lemma

Those squares are universal for "-equivalence.

Corollary

f W iff we have factorizations:

C D f Pⁿ

Oⁿ jn

in

!Oⁿ

Oⁿ (solution set condition)

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Conclusion

We have proved the following properties:

#

I-fib W;

#

W has the 3 for 2 property;

#

I-cof W is closed under pushout, retract, and tranfinite composition;

#

I and W satisfy the solution set condition.

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References

Y. Lafont, François Métayer & Krzysytof Worytkiewicz, A folk model structure on omega-cat (ArXiv, 2007) Y. Lafont & François Métayer, Polygraphic resolutions and homology of monoids (submitted)

François Métayer, Cofibrant objects among higher dimensional categories (HHA, 2008)

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