Simplicial acyclic models Michael Barr Math & Stats, McGill University Dedicated to the memory of Charles Wells
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 operators except d 0 . 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. 2 / 20
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 d 0 . 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. 3 / 20
The classical AM theorem Let me begin by giving a slightly modified version (due, in fact, to Guetta) of the classical AM theorem. � 0 and L • � 0 be chain � K − 1 � L − 1 Theorem. Suppose K • complexes in an abelian category with an augmented functor ( G , ǫ ) . Assume that for each n ≥ 0 , there is a presentation � GK n such that ǫ K n .θ n = id and that the mapping θ n : K n � GL − 1 � 0 is contractible. Then any augmented complex GL • � L − 1 extends to a chain map f • : K • � L • and any f − 1 : K − 1 two extensions are homotopic. The proof is by induction. Let t be the contracting homotopy and define f n as the composite Gf n − 1 θ n ǫ L n d t � GK n � GK n − 1 � GL n − 1 � GL n � L n K n 4 / 20
Kleisli’s theorem � X − 1 and Y • � Y − 1 be augmented face Theorem. Let X • complexes in a category C . Assume that for each n ≥ 0 , there is a � GX n such that ǫ X n .θ n = id and that presentation map θ n : X n Gd i .θ n = θ n − 1 d i for 1 ≤ i ≤ n and that GY • � GY − 1 is acyclic. � Y − 1 extends to a semisimplicial map Then any f − 1 : X − 1 � Y • . Moreover, any two extensions are equivalent with f • : X • respect to the equivalence relation generated by homotopy. The argument is similar to the above, with d 0 replacing d : Gf n − 1 θ n d 0 ǫ Y n t � GX n � GX n − 1 � GY n − 1 � GY n � Y n X n 5 / 20
(Semi)simplicial objects A semisimplicial object in a category C is a sequence � X n − 1 . . . X n , X n − 1 , . . . X 1 , X 0 equipped with morphisms d i n : X n for all n > 0 and 0 ≤ i ≤ n that satisfy the equations n = d j − 1 n − 1 d j d i n − 1 d i n whenever i < j . The d i n are called face operators. A simplicial object is a semisimplicial object that has, in addition, � X n +1 for all n ≥ 0 and 0 ≤ i ≤ n called bf morphisms s i n : X n degeneracies. They must satisfy the equations: n +1 s j n = s j n +1 s i − 1 1. s i for 0 ≤ j < i ≤ n + 1; n n +1 s j n = s j − 1 2. d i n − 1 d i n for 0 ≤ i < j ≤ n ; n = d i +1 3. d i n +1 s i n +1 s i n = id for 0 ≤ i ≤ n ; n +1 s j n = s j 4. d i n − 1 d i − 1 for 0 ≤ j < i − 1 ≤ n . n 6 / 20
(Semi)simplicial objects A semisimplicial object in a category C is a sequence � X n − 1 . . . X n , X n − 1 , . . . X 1 , X 0 equipped with morphisms d i n : X n for all n > 0 and 0 ≤ i ≤ n that satisfy the equations n = d j − 1 n − 1 d j d i n − 1 d i n whenever i < j . The d i n are called face operators. A simplicial object is a semisimplicial object that has, in addition, � X n +1 for all n ≥ 0 and 0 ≤ i ≤ n called bf morphisms s i n : X n degeneracies. They must satisfy the equations: n +1 s j n = s j n +1 s i − 1 1. s i for 0 ≤ j < i ≤ n + 1; n n +1 s j n = s j − 1 2. d i n − 1 d i n for 0 ≤ i < j ≤ n ; n = d i +1 3. d i n +1 s i n +1 s i n = id for 0 ≤ i ≤ n ; n +1 s j n = s j 4. d i n − 1 d i − 1 for 0 ≤ j < i − 1 ≤ n . n 6 / 20
(Semi)simplicial objects A semisimplicial object in a category C is a sequence � X n − 1 . . . X n , X n − 1 , . . . X 1 , X 0 equipped with morphisms d i n : X n for all n > 0 and 0 ≤ i ≤ n that satisfy the equations n = d j − 1 n − 1 d j d i n − 1 d i n whenever i < j . The d i n are called face operators. A simplicial object is a semisimplicial object that has, in addition, � X n +1 for all n ≥ 0 and 0 ≤ i ≤ n called bf morphisms s i n : X n degeneracies. They must satisfy the equations: n +1 s j n = s j n +1 s i − 1 1. s i for 0 ≤ j < i ≤ n + 1; n n +1 s j n = s j − 1 2. d i n − 1 d i n for 0 ≤ i < j ≤ n ; n = d i +1 3. d i n +1 s i n +1 s i n = id for 0 ≤ i ≤ n ; n +1 s j n = s j 4. d i n − 1 d i − 1 for 0 ≤ j < i − 1 ≤ n . n 6 / 20
(Semi)simplicial objects A semisimplicial object in a category C is a sequence � X n − 1 . . . X n , X n − 1 , . . . X 1 , X 0 equipped with morphisms d i n : X n for all n > 0 and 0 ≤ i ≤ n that satisfy the equations n = d j − 1 n − 1 d j d i n − 1 d i n whenever i < j . The d i n are called face operators. A simplicial object is a semisimplicial object that has, in addition, � X n +1 for all n ≥ 0 and 0 ≤ i ≤ n called bf morphisms s i n : X n degeneracies. They must satisfy the equations: n +1 s j n = s j n +1 s i − 1 1. s i for 0 ≤ j < i ≤ n + 1; n n +1 s j n = s j − 1 2. d i n − 1 d i n for 0 ≤ i < j ≤ n ; n = d i +1 3. d i n +1 s i n +1 s i n = id for 0 ≤ i ≤ n ; n +1 s j n = s j 4. d i n − 1 d i − 1 for 0 ≤ j < i − 1 ≤ n . n 6 / 20
� � Maps and homotopies From now on we will generally omit the lower indices on the face and degeneracy operators. � Y • is a If X • and Y • are (semi)simplicial objects a map f • : X • � Y n for all n ≥ 0 that commute sequence of morphisms f n : X n in the obvious way with all face and degeneracy operators. � Y • . A homotopy If f • and g • are simplicial maps from X • � Y n +1 for all n ≥ 0 and g • consists of maps h i h : f • n : X n 0 ≤ i ≤ n satisfying the following conditions (omit the last two in the semisimplicial case): 7 / 20
� � Homotopy equations 1. d 0 n +1 h 0 n = f n ; 2. d n +1 n +1 h n n = g n ; 3. d i +1 = d i +1 n +1 h i +1 n +1 h i n +1 for 0 ≤ i < n ; n n = h j − 1 n +1 h j 4. d i n − 1 d i n for 0 ≤ i < j ≤ n ; n +1 h j n = h j 5. d i n − 1 d i − 1 for 0 ≤ j < i − 1 ≤ n ; n n +1 h j n = h j n +1 s i − 1 6. s i for 0 ≤ j < i ≤ n + 1; n n +1 h j n = h j +1 7. s i n +1 s i 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 • . 8 / 20
� � Homotopy equations 1. d 0 n +1 h 0 n = f n ; 2. d n +1 n +1 h n n = g n ; 3. d i +1 = d i +1 n +1 h i +1 n +1 h i n +1 for 0 ≤ i < n ; n n = h j − 1 n +1 h j 4. d i n − 1 d i n for 0 ≤ i < j ≤ n ; n +1 h j n = h j 5. d i n − 1 d i − 1 for 0 ≤ j < i − 1 ≤ n ; n n +1 h j n = h j n +1 s i − 1 6. s i for 0 ≤ j < i ≤ n + 1; n n +1 h j n = h j +1 7. s i n +1 s i 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 • . 8 / 20
� � Homotopy equations 1. d 0 n +1 h 0 n = f n ; 2. d n +1 n +1 h n n = g n ; 3. d i +1 = d i +1 n +1 h i +1 n +1 h i n +1 for 0 ≤ i < n ; n n = h j − 1 n +1 h j 4. d i n − 1 d i n for 0 ≤ i < j ≤ n ; n +1 h j n = h j 5. d i n − 1 d i − 1 for 0 ≤ j < i − 1 ≤ n ; n n +1 h j n = h j n +1 s i − 1 6. s i for 0 ≤ j < i ≤ n + 1; n n +1 h j n = h j +1 7. s i n +1 s i 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 • . 8 / 20
Augmented (semi)simplicial object; contracting homotopy If X • is a (semi)simplicial object, an augmentation is a map � X − 1 such that d 0 d 0 0 d 0 1 = d 0 0 d 1 0 : X 0 1 . Note that this just continues the conditions on face operators one more degree. We � X − 1 such an augmented simplicial object. will denote by X • � X − 1 is an augmented simplicial object a contraction is a If X • � X n +1 for n ≥ − 1 that satisfy sequence of maps t n : X n 1. d 0 t = id ; 2. d i t = td i − 1 for i > 0; 3. s 0 t = tt ; 4. s i t = ts i − i for i > 0. Note that these equations are exactly the equations that a new degeneracy labeled s − 1 would satisfy. 9 / 20
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