Simplicial acyclic models Michael Barr Math & Stats, McGill - - PowerPoint PPT Presentation

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Simplicial acyclic models

Michael Barr

Math & Stats, McGill University

Dedicated to the memory of Charles Wells

SLIDE 2

Abstract

In 1974 Kleisli published a paper on acyclic models for semi-simplicial complexes (also known as face complexes). This differed from the theorem for chain complexes in that the “presentation” mapping was required to commute with all face

• perators except d0. I extend this to simplicial complexes, adding

that the presentation commute with all degeneracies. I also show that the standard resolution of any cotriple satisfies these conditions with respect to the cotriple.

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L´ eonard Guetta’s question

Maybe two months ago, I started a correspondence with a student in France named L´ eonard Guetta. It started with some comments and perspectives on the classical AM theorem for chain complexes. But at some point he asked if there was an AM theorem for simplicial objects in a category. I dimly recalled having seen such a theorem—it turned out to be Kleisli’s paper. What I recalled was the assumption that the presentation mapping commuted with all the face operators except d0. So I sat down to prove such a

• theorem. To get it for simplicial objects I had to suppose also that

the presentation mapping commuted with all the degeneracies.

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SLIDE 4

The classical AM theorem

Let me begin by giving a slightly modified version (due, in fact, to Guetta) of the classical AM theorem.

• Theorem. Suppose K•

K−1 0 and L• L−1 0 be chain

complexes in an abelian category with an augmented functor (G, ǫ). Assume that for each n ≥ 0, there is a presentation mapping θn : Kn

GKn such that ǫKn.θn = id and that the

augmented complex GL•

GL−1 0 is contractible. Then any

f−1 : K−1

L−1 extends to a chain map f• : K• L• and any

two extensions are homotopic. The proof is by induction. Let t be the contracting homotopy and define fn as the composite Kn

θn

GKn

d

GKn−1

Gfn−1

GLn−1

t

GLn

ǫLn

Ln

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SLIDE 5

Kleisli’s theorem

• Theorem. Let X•

X−1 and Y• Y−1 be augmented face

complexes in a category C. Assume that for each n ≥ 0, there is a presentation map θn : Xn

GXn such that ǫXn.θn = id and that

Gdi.θn = θn−1di for 1 ≤ i ≤ n and that GY•

GY−1 is acyclic.

Then any f−1 : X−1

Y−1 extends to a semisimplicial map

f• : X•

Y•. Moreover, any two extensions are equivalent with

respect to the equivalence relation generated by homotopy. The argument is similar to the above, with d0 replacing d: Xn

θn

GXn

d0

GXn−1

Gfn−1

GYn−1

t

GYn

ǫYn

Yn

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SLIDE 6

(Semi)simplicial objects

A semisimplicial object in a category C is a sequence . . . Xn, Xn−1, . . . X1, X0 equipped with morphisms di

n : Xn

Xn−1

for all n > 0 and 0 ≤ i ≤ n that satisfy the equations di

n−1dj n = dj−1 n−1di n whenever i < j. The di n are called face

• perators.

A simplicial object is a semisimplicial object that has, in addition, morphisms si

n : Xn

Xn+1 for all n ≥ 0 and 0 ≤ i ≤ n called bf

• degeneracies. They must satisfy the equations:
• 1. si

n+1sj n = sj n+1si−1 n

for 0 ≤ j < i ≤ n + 1;

• 2. di

n+1sj n = sj−1 n−1di n for 0 ≤ i < j ≤ n;

• 3. di

n+1si n = di+1 n+1si n = id for 0 ≤ i ≤ n;

• 4. di

n+1sj n = sj n−1di−1 n

for 0 ≤ j < i − 1 ≤ n.

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SLIDE 7

(Semi)simplicial objects

A semisimplicial object in a category C is a sequence . . . Xn, Xn−1, . . . X1, X0 equipped with morphisms di

n : Xn

Xn−1

for all n > 0 and 0 ≤ i ≤ n that satisfy the equations di

n−1dj n = dj−1 n−1di n whenever i < j. The di n are called face

• perators.

A simplicial object is a semisimplicial object that has, in addition, morphisms si

n : Xn

Xn+1 for all n ≥ 0 and 0 ≤ i ≤ n called bf

• degeneracies. They must satisfy the equations:
• 1. si

n+1sj n = sj n+1si−1 n

for 0 ≤ j < i ≤ n + 1;

• 2. di

n+1sj n = sj−1 n−1di n for 0 ≤ i < j ≤ n;

• 3. di

n+1si n = di+1 n+1si n = id for 0 ≤ i ≤ n;

• 4. di

n+1sj n = sj n−1di−1 n

for 0 ≤ j < i − 1 ≤ n.

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SLIDE 8

(Semi)simplicial objects

A semisimplicial object in a category C is a sequence . . . Xn, Xn−1, . . . X1, X0 equipped with morphisms di

n : Xn

Xn−1

for all n > 0 and 0 ≤ i ≤ n that satisfy the equations di

n−1dj n = dj−1 n−1di n whenever i < j. The di n are called face

• perators.

A simplicial object is a semisimplicial object that has, in addition, morphisms si

n : Xn

Xn+1 for all n ≥ 0 and 0 ≤ i ≤ n called bf

• degeneracies. They must satisfy the equations:
• 1. si

n+1sj n = sj n+1si−1 n

for 0 ≤ j < i ≤ n + 1;

• 2. di

n+1sj n = sj−1 n−1di n for 0 ≤ i < j ≤ n;

• 3. di

n+1si n = di+1 n+1si n = id for 0 ≤ i ≤ n;

• 4. di

n+1sj n = sj n−1di−1 n

for 0 ≤ j < i − 1 ≤ n.

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SLIDE 9

(Semi)simplicial objects

A semisimplicial object in a category C is a sequence . . . Xn, Xn−1, . . . X1, X0 equipped with morphisms di

n : Xn

Xn−1

for all n > 0 and 0 ≤ i ≤ n that satisfy the equations di

n−1dj n = dj−1 n−1di n whenever i < j. The di n are called face

• perators.

A simplicial object is a semisimplicial object that has, in addition, morphisms si

n : Xn

Xn+1 for all n ≥ 0 and 0 ≤ i ≤ n called bf

• degeneracies. They must satisfy the equations:
• 1. si

n+1sj n = sj n+1si−1 n

for 0 ≤ j < i ≤ n + 1;

• 2. di

n+1sj n = sj−1 n−1di n for 0 ≤ i < j ≤ n;

• 3. di

n+1si n = di+1 n+1si n = id for 0 ≤ i ≤ n;

• 4. di

n+1sj n = sj n−1di−1 n

for 0 ≤ j < i − 1 ≤ n.

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SLIDE 10

Maps and homotopies

From now on we will generally omit the lower indices on the face and degeneracy operators. If X• and Y• are (semi)simplicial objects a map f• : X•

Y• is a

sequence of morphisms fn : Xn

Yn for all n ≥ 0 that commute

in the obvious way with all face and degeneracy operators. If f• and g• are simplicial maps from X•

Y•. A homotopy

h : f•

• g• consists of maps hi

n : Xn

Yn+1 for all n ≥ 0 and

0 ≤ i ≤ n satisfying the following conditions (omit the last two in the semisimplicial case):

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SLIDE 11

Homotopy equations

• 1. d0

n+1h0 n = fn;

• 2. dn+1

n+1hn n = gn;

• 3. di+1

n+1hi+1 n

= di+1

n+1hi n+1 for 0 ≤ i < n;

• 4. di

n+1hj n = hj−1 n−1di n for 0 ≤ i < j ≤ n;

• 5. di

n+1hj n = hj n−1di−1 n

for 0 ≤ j < i − 1 ≤ n;

• 6. si

n+1hj n = hj n+1si−1 n

for 0 ≤ j < i ≤ n + 1;

• 7. si

n+1hj n = hj+1 n+1si n for 0 ≤ i ≤ j ≤ n.

Homotopy is not symmetric nor transitive and, in the semisimplicial case, not even reflexive. We denote the equivalence relation generated by

• by ∼ and say that f• is

wide homotopic to g• when f• ∼ g•.

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SLIDE 12

Homotopy equations

• 1. d0

n+1h0 n = fn;

• 2. dn+1

n+1hn n = gn;

• 3. di+1

n+1hi+1 n

= di+1

n+1hi n+1 for 0 ≤ i < n;

• 4. di

n+1hj n = hj−1 n−1di n for 0 ≤ i < j ≤ n;

• 5. di

n+1hj n = hj n−1di−1 n

for 0 ≤ j < i − 1 ≤ n;

• 6. si

n+1hj n = hj n+1si−1 n

for 0 ≤ j < i ≤ n + 1;

• 7. si

n+1hj n = hj+1 n+1si n for 0 ≤ i ≤ j ≤ n.

Homotopy is not symmetric nor transitive and, in the semisimplicial case, not even reflexive. We denote the equivalence relation generated by

• by ∼ and say that f• is

wide homotopic to g• when f• ∼ g•.

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SLIDE 13

Homotopy equations

• 1. d0

n+1h0 n = fn;

• 2. dn+1

n+1hn n = gn;

• 3. di+1

n+1hi+1 n

= di+1

n+1hi n+1 for 0 ≤ i < n;

• 4. di

n+1hj n = hj−1 n−1di n for 0 ≤ i < j ≤ n;

• 5. di

n+1hj n = hj n−1di−1 n

for 0 ≤ j < i − 1 ≤ n;

• 6. si

n+1hj n = hj n+1si−1 n

for 0 ≤ j < i ≤ n + 1;

• 7. si

n+1hj n = hj+1 n+1si n for 0 ≤ i ≤ j ≤ n.

Homotopy is not symmetric nor transitive and, in the semisimplicial case, not even reflexive. We denote the equivalence relation generated by

• by ∼ and say that f• is

wide homotopic to g• when f• ∼ g•.

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SLIDE 14

Augmented (semi)simplicial object; contracting homotopy

If X• is a (semi)simplicial object, an augmentation is a map d0

0 : X0

X−1 such that d0

0d0 1 = d0 0d1

• 1. Note that this just

continues the conditions on face operators one more degree. We will denote by X•

X−1 such an augmented simplicial object.

If X•

X−1 is an augmented simplicial object a contraction is a

sequence of maps tn : Xn

Xn+1 for n ≥ −1 that satisfy

• 1. d0t = id;
• 2. dit = tdi−1 for i > 0;
• 3. s0t = tt;
• 4. sit = tsi−i for i > 0.

Note that these equations are exactly the equations that a new degeneracy labeled s−1 would satisfy.

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SLIDE 15

Augmented (semi)simplicial object; contracting homotopy

If X• is a (semi)simplicial object, an augmentation is a map d0

0 : X0

X−1 such that d0

0d0 1 = d0 0d1

• 1. Note that this just

continues the conditions on face operators one more degree. We will denote by X•

X−1 such an augmented simplicial object.

If X•

X−1 is an augmented simplicial object a contraction is a

sequence of maps tn : Xn

Xn+1 for n ≥ −1 that satisfy

• 1. d0t = id;
• 2. dit = tdi−1 for i > 0;
• 3. s0t = tt;
• 4. sit = tsi−i for i > 0.

Note that these equations are exactly the equations that a new degeneracy labeled s−1 would satisfy.

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SLIDE 16

Augmented (semi)simplicial object; contracting homotopy

If X• is a (semi)simplicial object, an augmentation is a map d0

0 : X0

X−1 such that d0

0d0 1 = d0 0d1

• 1. Note that this just

continues the conditions on face operators one more degree. We will denote by X•

X−1 such an augmented simplicial object.

If X•

X−1 is an augmented simplicial object a contraction is a

sequence of maps tn : Xn

Xn+1 for n ≥ −1 that satisfy

• 1. d0t = id;
• 2. dit = tdi−1 for i > 0;
• 3. s0t = tt;
• 4. sit = tsi−i for i > 0.

Note that these equations are exactly the equations that a new degeneracy labeled s−1 would satisfy.

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SLIDE 17

Augmented (semi)simplicial object; contracting homotopy

If X• is a (semi)simplicial object, an augmentation is a map d0

0 : X0

X−1 such that d0

0d0 1 = d0 0d1

• 1. Note that this just

continues the conditions on face operators one more degree. We will denote by X•

X−1 such an augmented simplicial object.

If X•

X−1 is an augmented simplicial object a contraction is a

sequence of maps tn : Xn

Xn+1 for n ≥ −1 that satisfy

• 1. d0t = id;
• 2. dit = tdi−1 for i > 0;
• 3. s0t = tt;
• 4. sit = tsi−i for i > 0.

Note that these equations are exactly the equations that a new degeneracy labeled s−1 would satisfy.

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SLIDE 18

Augmented (semi)simplicial object; contracting homotopy

If X• is a (semi)simplicial object, an augmentation is a map d0

0 : X0

X−1 such that d0

0d0 1 = d0 0d1

• 1. Note that this just

continues the conditions on face operators one more degree. We will denote by X•

X−1 such an augmented simplicial object.

If X•

X−1 is an augmented simplicial object a contraction is a

sequence of maps tn : Xn

Xn+1 for n ≥ −1 that satisfy

• 1. d0t = id;
• 2. dit = tdi−1 for i > 0;
• 3. s0t = tt;
• 4. sit = tsi−i for i > 0.

Note that these equations are exactly the equations that a new degeneracy labeled s−1 would satisfy.

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SLIDE 19

The main theorem

• Theorem. Let X•

X−1 and Y• Y−1 be augmented simplicial

• bjects of C and (G, ǫ) be an augmented endofunctor. Suppose for

all n ≥ 0, there is a presentation θn : Xn

GXn that satisfies

ǫXn.θn = id, Gdi.θn = θn−1di for all 0 < i ≤ n, and Gsi.θn = θn+1si for all 0 ≤ i ≤ n. Suppose also that GY•

GY−1 is contractible. Then any f−1 : X−1 Y−1 extends

to a map λ•(f−1) : X•

Y•. Any two extensions of f−1 are wide

homotopic. We will call λ•(f−1) the lifting of f−1. The reason for introducing this notation is that what we will actually show is that for any extension f• of f−1, we have f•

• λ•(f−1). Therefore if g• is

another lifting we also have g•

• λ(f−1) and therefore f• ∼ g•.

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SLIDE 20

The construction

The construction begins the same way as Kleisli’s. Write f = f−1 and defined λn(f ) inductively as the composite Xn

θn

GXn

d0

GXn−1

Gλn−1(f )

GYn−1

t

GYn

ǫYn

Yn

The proof that diλn(f )λn−1di depends on whether i = 0 or i > 0. In the first case, we have Xn GXn

θn GXn

GXn−1

Gd0 GXn−1

GYn−1

Gλn−1(f )

GYn−1

GYn

t

GYn

Yn

ǫYn

• Xn

Xn

=

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹

GXn Xn

ǫXn

• GXn−1

Xn−1

ǫXn−1

• GYn−1

GYn−1

=

• ❊

❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊

GYn−1 Yn−1

ǫYn−1

• ❘

❘ ❘ ❘ ❘

GYn GYn−1

Gd0

• Yn

Yn−1

d0

• Xn

Xn−1

d0 Xn−1

Yn−1

λn−1(f )

• 11 / 20

SLIDE 21

Main theorem, cont’d

For i > 0: Xn GXn

θn GXn

GXn−1

Gd0 GXn−1

GYn−1

Gλn−1(f )

GYn−1

GYn

t

GYn

Yn

ǫYn

• Xn−1

GXn−1

θn−1

GXn−1

GXn−2

Gd0

GXn−2

GYn−2

Gλn−2(f )

GYn−2

GYn−1

t GYn−1

Yn−1

ǫYn−1

Xn Xn−1

di

• GXn

GXn−1

Gdi

• GXn−1

GXn−2

Gdi−1

• GYn−1

GYn−2

Gdi−1

• GYn

GYn−1

Gdi

• Yn

Yn−1

di

• This is a constant theme in the proof of the theorem; thecase that

an index is 0 is separate from the one in which it is positive. I will not here give all the necessary diagrams; they can be found in the paper.

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SLIDE 22

The homotopy

I claim that if f• extends f−1, then there is a homotopy h : f•

• λ•(f−1). We define the homotopy inductively as follows.

For any n ≥ 0 let h0 = h0

n : Xn

Yn+1 be the composite

Xn

θn

GXn

Gfn

GYn

t

GYn+1

ǫYn+1

Yn+1

while for 1 ≤ i ≤ n, we define hi as the composite Xn

θn

GXn

Gd0

GXn−1

Ghi−1

GYn

t

GYn+1

ǫYn+1

Yn+1

These definitions are the same as Kleisli’s and the first five equations in the definition of homotopy work exactly as in his case. The nasty ones, involving the interaction between the homotopy and the degeneracies, are the last two, especially the last. It says that for 0 ≤ i ≤ j ≤ n, sihj = hj+1si. There are two special cases to consider. The first one is that i = j = 0 and the second that 0 = i < j.

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SLIDE 23

Homotopy cont’d

Let i = j = 0. The diagram GXn G 2Xn

Gθn

G 2Xn

G 2Yn

G 2fn

G 2Yn

G 2Yn+1

Gt

• GXn

GXn

=

• ❍

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

G 2Xn GXn

GǫXn

• G 2Yn

GYn

GǫYn

• G 2Yn+1

GYn+1

GǫYn+1

• GXn

GYn

Gfn

GYn

GYn+1

t

• shows that Gh0 = t.Gfn. Then we have

Xn GXn

θn

• Xn

Xn+1

s0

✎✎✎✎✎✎✎✎✎

GXn GXn+1

Gs0

✎✎✎✎✎✎✎✎✎

GXn GYn

Gfn

GYn

GYn+1

t

GYn+1

Yn+1

ǫYn+1

GXn GXn

=

• ✴

✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴

GXn GYn+1

Gh0

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

GYn GYn+1

t

• GYn+1

GYn+2

Gs0

• Yn+1

Yn+2

s0

• Xn+1

GXn+1

θn+1

GXn+1

GXn

Gd0 GXn

GYn+1

Gh0 GYn+1

GYn+2

t GYn+2

Yn+2

ǫYn+2

• 14 / 20

SLIDE 24

Homotopy, cont’d

The diagram GXn G 2Xn

Gθn G 2Xn

G 2Xn−1

G 2d0 G 2Xn−1

G 2Yn

G 2hj−1

G 2Yn

G 2Yn+1

Gt

GXn GXn

=

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄

G 2Xn GXn

GǫXn

• G 2Xn−1

GXn−1

GǫXn−1

• G 2Yn

GYn

GǫYn

• G 2Yn+1

GYn+1

GǫYn+1

• GXn

GXn−1

Gd0 GXn−1

GYn

Ghj−1 GYn

GYn+1

t

• shows that Ghj = t.Ghj−1.Gd0. Then we have

Xn GXn

θn

GXn

GXn−1

Gd0 GXn−1

GYn

Ghj−1 GYn

GYn+1

t

GYn+1

Yn+1

ǫYn+1

GXn GYn+1

Ghj

• ❖

❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖

GYn GYn+1

t

• GYn+1

GYn+2

Gs0

• Yn+1

Yn+2

s0

• Xn

Xn+1

s0

• Xn+1

GXn+1

θn+1

GXn+1

GXn

Gd0 GXn

GYn+1

Ghj

GXn GXn+1

Gs0

• GXn

GXn

=

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄

GYn+1 GYn+2

t GYn+2

Yn+2

ǫYn+2

• 15 / 20

SLIDE 25

Homotopy, conclusion

Finally, we consider the case that 0 < i ≤ j: Xn GXn

θn

GXn

GXn−1

Gd0 GXn−1

GYn

Ghj−1 GYn

GYn+1

t

GYn+1

Yn+1

ǫYn−1

Xn Xn+1

si

• GXn

GXn+1

Gsi

• GXn−1

GXn

Gsi−1

• GYn

GYn+1

Gsi−1

• GYn+1

GYn+2

Gsi

• Yn+1

Yn+2

si

• Xn+1

GXn+1

θn+1 GXn+1

GXn

Gd0 GXn

GYn+1

Ghj GYn+1

GYn+2

t GYn+2

Yn+2

ǫYn+2

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SLIDE 26

Cotriple resolutions

We now look at the case that G = (G, ǫ, δ) is a cotriple on C. The cotriple resolution of an object c ∈ C is the simplicial complex consisting of the object G n+1C in degree n with face operators di = G iǫG n−1 and degeneracy operators si = G iδG n−i. I claim that the cotriple resolution of every object satisfies both the presentability and acyclicity conditions of the main theorem. The

• bject C sits in degree −1 with face operator ǫ : GC

C.

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SLIDE 27

cotriple resolutions, proof

We begin by establishing four identities that we need. CR-1 GǫG n.δG n = id; CR-2 For i > 0, G i+1ǫ G n−i.δG nC = δG iǫG n−i; CR-3 GδG n.δG n = δG n+1.δG n; CR-4 For i > 0, G i+1δ.G n−i = δG n+1.G iδG n−i.

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SLIDE 28

cotriple resolutions, proof, completed

CR-1 is just the cotriple identity Gǫ.δ = id, applied to G n. CR-2 is naurality of δ, applied as follows: G 2(G i−1ǫG n−i).δG n = δG n−1.G(G i−1ǫG n−i). CR-3 is the cotriple identity Gδ.δ = δG.δ applied to G n. CR-4 is naturality of δ applied as follows: G 2(G i−1δG n−i).δG n = δG n+1.G(G i−1δG n−i).

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SLIDE 29

Contractible simplicial objects

A few weeks ago, I started to raise the question on Math Overflow

• f when a simplicial object in a category is contractible. I

discovered that the question had been posed—but not answered—some years earlier. I was a bit surprised to find that there is a more-or-less obvious answer, at least when the ambient category has split idempotents. Suppose X• is a simplicial object and there are maps t = tn : Xn

Xn+1 for all n ≥ 0 such that

d0t = id, dit = tdi−1 for i > 0, ts0 = tt, and tsi = si−1t for i > 0. We have, in X0 d1td1t = d1d2tt = d1d1tt = d1td0t = d1t so that d1t is idempotent. If we factor it as X0

d0

X−1

t−1

X0

we nearly have a contractible augmented simplicial object. The

• nly thing to verify is that d0d1 = d0d0. But

td0d0 = d1td0 = d1d1t = d1d2t = d1td1 = td0d1 and t being a (split) monic can be canceled to give d0d0 = d0d1.

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