Delooping of the $ K $-Theory of Strictly Derivable Waldhausen Categories

Delooping of the K-theory of strictly derivable Waldhausen categories

Satoshi Mochizuki

Introduction

In this short note, for a morphism of Waldhausen categories f : A = (A ,wA) → B = (B,wB), we will define Cone f to be a Waldhausen category. There exists the canonical morphism of Waldhausen categories κ f : B → f κ f Cone f (see Definition 2.5). We will show that the sequence A → B → Cone f induces fibration sequence of K( f ) K(κ f ) spaces K(A) → K(B) → K(Cone f ) on connective K-theory (see Theorem 2.7). Moreover we will define a notion of strictly derivable Waldhausen categories (see Definition 1.6) and define non-connective K-theory for strictly derivable Waldhausen categories (see Definition 3.12).

1 Admissible classes of morphisms and w-thick subcategories

In this section, let C be a category with coﬁbrations. We ﬁx a zero object 0 of C .

Deﬁnition 1.1 (Admissible class of morphisms). We say that a class w of morphisms in C is admissible if w satisﬁes the following three conditions. (Closed under isomorphisms). w contains all isomorphisms.

(Two out of three property for compositions). w satisfies the two out of three property. Namely for a f g pair of composable morphisms x → y → z, if two of g f , f and g are in w, then the third one is also in w. (Two out of three property for cofibration sequences). In the commutative diagram of cofibration sequences below j q x / / y / / y/x (1)

a b c arXiv:1803.07254v2 [math.KT] 6 Aug 2020 x0 / / y0 / / y0/x0, j0 q0

if two of a, b and c are in w, then the third one is also in w.

Lemma 1.2. Assume that C is enriched over the category of abelian groups. If w is an admissible class of morphisms in C , then w satisﬁes the gluing axiom of [Wal85]. In particular the pair (C ,w) is a category with coﬁbrations and weak equivalences.

Proof. Since C is close under ﬁnite coproducts, C is an additive category. In the commutative diagram below, assume that a, b and c are in w. y o x / / z

b a c y0 o x0 / / z0.

1 Then there are commutative diagrams of coﬁbrations sequences.

y / / y ⊕ z / / z x / / y ⊕ z / / y tx z

b 0 b c a 0 bt c b 0 c 0 c a 0 0 0 0 0 0 0 0 0 y / / y ⊕ z / / z , x / / y ⊕ z / / y tx0 z .

b 0 By the left diagram above, it turns out that 0 c is in w and therefore b ta c is also in w by the right diagram above.

We fix w an admissible class of morphisms in C . Definition 1.3 (Thick subcategory, w-closed subcategory). Let S be a class of objects in C . We sometimes regard S as a full subcategory of C . We say that S is a left prethick subcategory of C if it contains the zero object 0 of C and if for a cofibration sequence x y y/x in C with y/x ∈ S , x is in S if and only if y is in S . We say that S is a right prethick subcategory of C if it contains the zero object 0 of C and if for a cofibration sequence x y y/x in C with x ∈ S , y is in S if and only if y/x is in S . We say that S is a prethick subcategory of C if it is both a left and right prethick subcategory of C . We say that S is left thick (respectively right thick, thick) if S is left prethick (respectively right prethick, prethick) and moreover if S is closed under retractions. The last condition means that for an object x in C i p if there exists morphisms x → y → x such that pi = idx and y is in S , then x is also in S . We say that S is w-closed if it contains the zero object 0 and if for an object x in C , if there exists an object y in S and if there exists a zig-zag sequence of morphisms in w which connects x and y, then x is also in S . For an admissible class u of morphisms in C , we write C u for the full subcategory of C consisting of those objects x such that the canonical morphism 0 → x is in u. Then by the lemma below, C u is a thick subcategory of C . We say that S is left w-prethick (respectively right w-prethick, right w-thick, w-prethick, w-thick) if it is left prethick (respectively left thick, right prethick, right thick, prethick, thick) and if it contains C w. Lemma 1.4. For an admissible class u of morphisms in C , C u is a thick subcategory of C . Moreover if u contains w, then C u is w-closed.

Proof. For a coﬁbration sequence x y y/x, by applying the axiom of admissible classes to the commutative diagram below, it turns out that if two of x, y and y/x are in C u, then the third one is also in C u.

0 / / 0 / / 0

x / / y / / y/x.

f Next let x → y be a morphism in w and assume that x (resp. y) is in C u, then y (resp. x) is also in C u by the two out of three property of u. f / x ^ @ y

The rest of this section, we assume that the pair (C ,w) satisﬁes the following factorization axiom in [Sch06, A.5].

f 1.5 (Factorization axiom). For any morphism x → y in C , there exists a coﬁbration i f : x u f and a morphism q f : u f → y in w such that f = q f i f . In this case, we call a triple (i f ,q f ,u f ) a (w-)factorization of f .

2 Recall that in [Cis10, 1.1], we say that a Waldhausen category is derivable if it satisfies the two out of three property for compositions and if it satisfies the factorization axiom. Thus we define the following notion. Definition 1.6 (Strictly derivable Waldhausen category). We say that a Waldhausen category (C ,w) is strictly derivable if it satisfies the factorization axiom and w is an admissible class. The following two lemmata are useful to construct the correspondence between admissible classes and prethick subcategories. Lemma 1.7. Let i x / y (2)

a b x0 / y0 i0 be a commutative square in C and let (ia,qq,ua) be a factorization of a. Then there exists a factorization 0 (ib,qb,ub) of b: y → y and a morphism j : ua → ub which makes the diagram below commutative and the canonical morphism ua tx y → ub induced from the universal property of pushout diagram is a coﬁbration.

i x / y (3) ia ib j ua / ub

qa qb y / y0. i0

Moreover assume that i is a cofibration. Then we can take j as a cofibration. 00 0 Proof. Let (i ,qb,ub) be a factorization of ua tx y → y the induced morphism from the universal property i00 i00 of pushout. We write ib and j for the compositions y ua tx y ub and ua → ua tx y ub respectively. Moreover if i is a cofibration, then the induced morphism ua ua tx y is a cofibration. Therefore j is a cofibration. Lemma 1.8. Let i p y o o x / z (4)

b a c y0 o o x / z0 i0 p0

0 be a commutative diagram in C with i and i coﬁbrations and let (ia,qa,ua), (ib,qb,ub) and (ic,qc,uc) be 00 00 factorizations of a, b and c respectively and let i : ua → ub and p : ua → uc be morphisms which makes diagram below commutative. i p y o o x / z (5)

ib ia ic i00 p00 ub o o ua / uc

qb qa qc y0 o o x / z0. i0 p0

3 Assume that ytx ua → ub the induced morphism from the universal property of pushout is a coﬁbration. Then the triple (ib tia ic,qb tqa qc,ub tua uc) is a factorization of b ta c.

Proof. ib tia ic is a coﬁbration by [Wal85, 1.1.2]. qb tqa qc is a morphism in wB by the gluing axiom. bta c = (qb tqa qc)(ib tia ic). Thus the triple (ib tia ic,qb tqa qc,ub tua uc) is a factorization of b ta c. Deﬁnition 1.9 (S -weak equivalences). Let S be a w-left thick subcategory of C . Then for a morphism f x → y in C , the following two conditions are equivalent by Lemma 1.10 below and in this case, we say that f is an (S ,w)-weak equivalence or simply S -weak equivalence.

• There exists a factorization (i f , p f ,u f ) of f such that u f /x is in S .

• For any factorization (i f , p f ,u f ) of f , u f /x is in S .

We denote the class of all (S ,w)-weak equivalences in C by wS ,C ,w or shortly wS . Lemma 1.10. Let S be a w-left thick subcategory of C . Then

i j (i) Let x y z be a pair of composable coﬁbrations in C such that z/y is in S . Then y/x is in S if and only if z/x is in S .

i w (ii) If x y is a coﬁbration and a morphism in w, then y/x is in C . (iii) In the commutative diagram below, assume that both i and i0 are coﬁbrations and both p and p0 are morphisms in w, i x / / z i0 p z0 / y, p0

then z/x is in S if and only if z0/x is in S .

Proof. (i) By considering a coﬁbrations sequence y/x z/x z/y, we obtain the result. (ii) By the commutative diagram below, it turns out that the canonical morphism y/x → 0 is in w by the gluing axiom. i x / / y / / y/x

i y y / 0.

Then since w satisﬁes two out of three property for compositions, it turns out that y/x is in C w ⊂ S . 0 0 (iii) By the factorization axiom, there exists a factorization (iptp0 ,qptp0 ,u) of p t p : z t z → y. Then by the i 0 two out of three property for w, the composition z → z t z0 p→tp u is a coﬁbration and a morphism in w. Thus by (ii), u/z is in S . Then by (i), z/x is in S if and only if u/x is in S . By symmetry, it is also equivalent to the condition that z0/x is in S . Proposition 1.11. Let S be a w-thick subcategory of C and let u be an admissible class of morphisms in C which contains w. Then

(i) wS is an admissible class of morphisms in C which contains w. (ii) We have the equalities C wS = S , (6)

wC u = u. (7)

4 Proof of Proposition 1.11. (i) For a morphism f : x → y in w, the triple (idx, f ,x) is a factorization of f such ∼ that x/x → 0 is in S . Thus f is in wS . In particular, wS contains all isomorphisms. We consider the commutative diagram of cofibration sequences (1). Let (ia,qa,ua) be a factorization of a. Then by Lemma 1.7 and Lemma 1.8, there exists factorizations (ib,qb,ub) and (ic,qc,uc) of b and c respectively and a cofibration sequence ua ub uc such that the induced morphisms ua tx y → ub is a cofibration and they make the diagram below commutative:

x / / y / / y/x

ia ib ic ua / / ub / / uc

qa qb qc x0 / / y0 / / (y/x).

Then the sequence ua/x ub/y uc/(y/x) is a cofibration sequence by [Wal85, 1.1.2]. Hence if two of ua/x, ub/y and uc/(y/x) are in S , then the third one is also in S . Namely if two of a, b and c are in wS , then the third one is also in wS . a b 0 Next let x → y → z be composable morphisms in C . Let (ia,qa,ua) be a factorization of a. Let (i ,qba,uba) 0 be a factorization of bqa : ua → z. Then we set iba := i ia and it turns out that the triple (iba,qba,uba) is a factorization of ba: x → z. By Lemma 1.7, there exists a factorization (ibatidy ,qbatidy ,ubatidy ) of batidy : xt y → z t y and a cofibration j : uba ubatidy such that uba tx (x t y) ubatidy the induced morphism from the universal property of pushout is a cofibration. Then by Lemma 1.8, there exists factorizations (ib,qb,ub), ji0 (i ,q ,u ) and (i ,q0 ,u ) of b, id and ba t id respectively and cofibration sequences u idy idy idy batidy batidy batidy y y a j ubatidy ub and uba ubatidy uidy such that they makes the diagrams below commutative: x / / x t y / / y x / / x t y / / y

ua / / ubatidy / / ub uba / / ubatidy / / uidy

y / / z t y / / z z / / z t y / / y.

By [Wal85, 1.1.2], the following sequences are coﬁbration sequences

ua/x ubatidy /(x t y) ub/y, (8)

uba/x ubatidy /(x t y) uidy /y. (9) w Since idy is a cofibration and a morphism in w, ub/y is in C ⊂ S by Lemma 1.10 (iii). Thus in the cofibration sequence (9), uba/x is in S if and only if ubatidy /(xty) is in S . Hence if two of ua/x, uba/x and ub/y are in S , then the third one is also in S . Namely wS satisfies the two out of three property for compositions. (ii) We show the equality (6). Let x be an object in C and let (i0,q0,u0) be a factorization of the canonical w morphism x → 0. Then since p0 : u0 → 0 is in w, u0 is in C ⊂ S . Thus in the cofibration sequence i0 x u0 u0/x, x is in S if and only if u0/x is in S . Thus x is in S if and only if x → 0 is in wS if and only if x is in C wS . f Next we prove the equality (7). Let x → y be a morphism in C and let (i f , p f ,u f ) be a factorization of f . u Then f is in wC u if and only if u f /x is in C . In the commutative diagram below,

i f x / / u f / / u f /x

f p f y y / 0,

5 f is in u if and only if the canonical morphism u f /x → 0 is in u and the last condition is equivalent to the condition that 0 → u f /x is in u by the two out of three property of u.

Corollary 1.12. Let S be a w-thick subcategory of C . Then S is w-closed.

wS Proof. Since wS contains w, C is w-closed by Lemma 1.4. By the equality (6), S is also w-closed.

2 Puppe ﬁbration sequences for connective K-theory

Let f : A = (A ,wA) → B = (B,wB) be a morphism of Waldhausen categories. We write Im f for the class ∼ of those objects x in B such that there exists an object y in A and an isomorphism x → f (y).

Deﬁnition 2.1 (Admissible morphisms). (i) We say that f is preadmissible if Im f is a wB-left thick subcategory (see Deﬁnition 1.3) of B.

(ii) We say that f is semi-admissible if it is preadmissible and for any cofibration i: f (x) f (y) in B, 0 0 there exists a cofibration i : x y in A such that f (i ) = i. (iii) We say that f is admissible if it satisfies the following four conditions hold.

(i) f is preadmissible. (ii) The association f : ObA → ObB on the classes of objects are injective. i p (iii) For any cofibration sequence f (x) f (y) f (z), there exists a unique cofibration sequence 0 p0 i 0 0 x y z such that f (i ) = i and f (p ) = p. Lemma 2.2. Let f : A → B be a semi-admissible morphism of Waldhausen categories. Then Im f is closed j under co-base change with respect to cofibrations. Namely let z x → y be morphisms with j cofibrations and x, y and z ∈ Im f . Then z tx y is also in Im f . In particular Im f is a wB-thick subcategory of B.

j0 0 0 Proof. Without loss of generality, we shall assume that there exists morphisms x z whose image of f is j x z. Then in the commutative diagram of coﬁbration sequences

j f (x0) / / f (z0) / / f (z0/x0)

0 0 z / / z tx y / / f (z /x ),

0 0 z and f (z /x ) are in Im f . Thus z tx y is also in Im f . The rest of this section, we assume that B is strictly derivable (see Deﬁnition 1.6) and f is preadmissible.

Deﬁnition 2.3. We write w f for wIm f the class of Im f -weak equivalences (see Deﬁnition 1.9). We call morphisms in w f f -weak equivalences. Lemma 2.4. Let f : A → B be a semiadmissible morphism of Waldhausen categories. Then

(i) w f is an admissible class of B which contains wB.

(ii) w f satisfies the gluing axiom. Proof. (i) follows from Proposition 1.11. 0 (ii) We consider the commutative diagram (4) in B with i and i cofibrations and a, b and c w f -weak equivalences. There exists (ia,qa,ua), (ib,qb,ub) and (ic,qc,uc) factorizations of a, b and c respectively 0 00 and a cofibration i : ua ub and a morphism p : ua → uc which makes the diagram (5) commutative

6 and y tx ua → ub the induced morphism is a coﬁbration by Lemma 2. Then by Lemma 1.8, the triple

(ib tia ic,qb tqa qc,ub tua uc) is a factorization of b ta c and the commutative square

ua/x / / ub/y

uc/z / / (uc tua ub)/(z tx y)

is a pushout diagram by [Wal85, 1.1.2.]. Since ua/x, ub/y and uc/z are in Im f , (uc tua ub)/(z tx y) is also in Im f by Lemma 2.2. Thus b ta c is in w f . Deﬁnition 2.5 (Cone of morphisms of Waldhausen categories). Let B be a strictly derivable Waldhausen category and let f : A → B be a semi-admissible morphism of Waldhausen categories. In this case, the pair (B,w f ) is a Waldhausen category by Lemma 2.4. We set Cone f := (B,w f ) and we denote the morphism of Waldhausen categories B → Cone f induced from idB : B → B by κ f : B → Cone f . We call Cone f the cone associated with f . By Lemma 2.4, we obtain the following.

Corollary 2.6. In the notation Deﬁnition 2.5, Cone f is a strictly derivable Waldhausen category. Theorem 2.7 (Puppe sequences for connective K-theory). Let B be a strictly derivable Waldhausen cate- f κ f gory and let f : A → B be an admissible morphism of Waldhausen categories. Then the sequence A → B → K( f ) K(κ f ) Cone f induces a ﬁbration sequence of spectra K(A) → K(B) → K(Cone f ) on K-theory.

[−] Proof. We define B(−wB) to be a simplicial subcategory of B by sending a totally ordered set [m] to [m] B(m,wB) where B(m,wB) is the full subcategory of B the functor category from [m] to B consisting of those functors x: [m] → B such that for any 0 ≤ i ≤ j ≤ m in [m], x(i ≤ j) is in wB. A morphism f : x → y in B(m,wB) is a level weak equivalence (respectively level cofibration) if fi : xi → yi is in wB (respectively a cofibration) for all 0 ≤ i ≤ m. Then we can make B(−,wB) into a simplicial Waldhausen category. We denote the class of all cofibrations in w f byw ¯ f . By [Wal85, 1.5.7.], there exists a fibration sequence

wAS· A → wBS· B → wS·S· f .

As in [Wal85, p.352], the third bisimplicial category wS·S· f can be identiﬁed withw ¯ f S· B(−,wB) by admis- sibility of wB and f and as in the original proof of ﬁbration sequences in [Wal85], there exists a commutative diagram of (bi)simplicial categories

wBS· B / w¯ f S· B(−,wB)

I w f S· B(−,wB) O II

wBS· B / w f S· B κ f where the maps I and II are homotopy equivalences by [Sch06, A.11.] and [Wal85, 1.6.5.] respectively. Hence we obtain the result.

3 Unbounded ﬁltered objects

In this section, let the pair C := (C ,w) be a strictly derivable Waldhausen category (see Deﬁnition 1.6).

7 Definition 3.1 (Filtered objects). Let N be the linearly ordered set of all natural numbers with the usual linear order. As usual we regard N as a category and we denote the category of functors and natural transformations from N to C by C N. We say that an object x in C N is a filtered object (in C ) if x(n ≤ n + 1) is a cofibration for all n ∈ N. We denote the full subcategory of C N consisting of filtered objects in C by F C . For a filtered x object x and for a natural number n, we write xn and in for an object x(n) in C and a morphism x(n ≤ n + 1) in C respectively. Let f : x → y be a morphism in F C . We say that f is a level weak equivalence if for each natural number n, fn is in w. We denote the class of all level weak equivalences by lwF C or simply lw. We say that f is a

Reedy cofibration if for each natural number n, fn and the induced morphism xn txn−1 yn−1 are cofibrations. We can make F C into a category of cofibrations by declaring the class of all Reedy cofibrations to be the class of cofibrations in F C (cf. [Wal85, 1.1.4.]). We will define several operations on F C .

Deﬁnition 3.2 (Truncation functor). For a natural number n, the inclusion functors F≥n C → F C and F≤n C → F C admit right adjoint functors σ≥n : F C → F≥n C and σ≤n : F C → F≤n C respectively. Ex- plicitly, for an object x and a morphism f : x → y in F C , we set

( ( x xk if k ≤ n σ≤nx ik if k ≤ n − 1 (σ≤nx)k := , ik := xn if k ≥ n idxn if k ≥ n

( ( x xk if k ≥ n σ≥nx ik if k ≥ n (σ≥nx) := , i := k 0 if k < n k 0 if k ≤ n − 1. ( ( fk if k < n fn if k ≥ n (σ≤n f )k := , (σ≥n f )k := fn if k ≥ n 0 if k ≤ n − 1.

There are adjunction morphisms σ≥nx → x and σ≤nx → x.

Definition 3.3 (Degree shift). Let n be a natural number and let f : x → y be a morphism in F C . We define x[n] and f [n]: x[n] → y[n] to be a filtered object in C and a morphism of filtered objects respectively by setting x[n] x x[n]k := xn+k and ik := in+k and f [n]k := fn+k. The association (−)[n]: F C → F C , x 7→ x[n] gives an exact functor. There exists a natural transformation θ : idF C → idF C [1]. Namely for a filtered object x in C , θx : x → x[1] x is defined by the formula θxk := ik for each natural number k. For a pair of natural number a < b, We write [a,b] θ : idF C [a] → idF C [b] for the compositions θ[b − 1]θ[b − 2]···θ[a + 1]θ[a]. For a full subcategory D of F C which is closed under the operation (−)[1]: D → D, we write ΘD for the family {θx : x → x[1]}x∈ObD of morphisms in D indexed by the class of objects in D.

Lemma 3.4. (i) lw is an admissible class of morphisms in F C . (ii) The pair (F C ,lw) satisfies the factorization axiom. Proof. A proof of assertion (i) is straightforward. We will give a proof of assertion (ii). Let n be a natural number and assume that σ≤n f : σ≤nx → σ≤ny admits a factorization (iσ≤n f , pσ≤n f ,uσ≤n f ). Then let a triple 0 y (i , p fn+1 ,u fn+1 ) be a factorization of a morphism fn+1 txn in p fn : xn+1 txn u fn → yn+1 and we denote the com- 0 0 i i u f positions xn+1 xn+1 txn u fn u fn+1 and u fn → xn+1 txn u fn u fn+1 by i fn+1 and in respectively. Then the pair of triples (iσ≤n f , pσ≤n f ,uσ≤n f ) and (i fn+1 , p fn+1 ,u fn+1 ) give a factorization of σ≤n+1 f : σ≤n+1x → σ≤n+1. By proceeding induction on n, we finally obtain a factorization of f . Definition 3.5 (Bounded and weakly bounded filtered objects). Let a ≤ b be a pair of natural numbers and let x be a filtered object in C . We say that x has amplitude contained in [a,b] if for any 0 ≤ k < a, xk = 0 and x for any b ≤ k, xk = xb and ik = idxb . In this case we write x∞ for xb = xb+1 = ···. Similarly for any morphism f : x → y in F C between objects which have amplitude contained in [a,b], we denote fb = fb+1 = ··· by f∞. We denote the full subcategory of F C consisting of those objects having amplitude contained in [a,b] by [ [ F[a,b] C . We also set Fb C := F[a,b] C and Fb,≥a C := F[a,b] C and call an object in Fb C a bounded a

8 filtered object (in C ). For a natural number n, we write F≤n C and F≥n C for the category F[0,n] C and the full subcategory of F C consisting of those objects x such that xk = 0 for all 0 ≤ k < n respectively. Let f : x → y be a morphism in C . We write j(x) and j( f ): j(x) → j(y) for the object and the morphism in F[0,0] C such that j(x)∞ = x and j( f )∞ = f . We define (−)∞ : Fb C → C and j: C → Fb C to be functors by sending an object x in Fb C to x∞ and sending an object x in C to j(x). We denote F[0,0] C by j(C ) and sometimes identify it with C via the functor j. x We say that a filtered object x in C is weakly bounded if there exists a natural number N such that in is in w for all n ≥ N. We write Fwb C for the full subcategory of F C consisting of all weakly bounded filtered objects. For a full subcategory D of F C , we denote the class of level weak equivalences in D by lw|D or lwD or simply lw. Definition 3.6 (Stable weak equivalences). A morphism f : x → y in F C is stably weak equivalence if there exists a natural number N such that fn is in w for all n ≥ N. For a full subcategory D of F C , we denote the class of all stably weak equivalences in D by wst|D or simply wst.

Lemma 3.7. (i) wst is an admissible class of morphisms in F C .

(ii) The exact functor j: C = (C ,w) → (Fb C ,wst) induces a homotopy equivalence K(C) → K(Fb C ;wst) of spectra on K-theory.

Proof. A proof of assertion (i) is straightforward. For (ii), notice that we have the equality (−)∞ · j = idC and there exists a natural weak equivalence idFb C → j·(−)∞ with respect to wst. Thus j induces a homotopy equivalence of spectra on K-theory.

Lemma 3.8. (i) wst|Fwb C is the smallest admissible class of morphisms in Fwb C which contains lw and ΘFwb C .

(ii) Fwb C is the smallest wst-prethick subcategory of F C which contains j(C ).

(iii) For a morphism f : x → y in Fwb C with x ∈ ObFb C , there is a factorization (i f , p f ,u f ) of f with respect to wst the class of stable weak equivalences such that u f is in Fb C .

Proof. (i) Let u be a class of morphisms in Fwb C which contains lw and ΘFwb C . Then we will show that u contains wst. First we will show that for any natural number N and for any object x in Fwb C , the natural [0,N] morphism σ≥Nx → x is in u. Notice that in the left commutative diagram below θσ≥N x : σ≥Nx → σ≥N[N], [0,N] θx : x → x[N] and idx[N] : σ≤Nx[N] → x[N] are in u. Thus the natural morphism σ≥Nx → x also in u by the two out of three property.

σ≥M f σ≥N x / x σ≥Mx / σ≥My

[0,N] [0,N] θσ≥N x θx (σ≥N x)[N] / x[N], x / y. f idx[N]

Next let f : x → y be a morphism in wst and let M be a natural number such that fn : xn → yn is in w for all n ≥ M. Then σ≥M f : σ≥Mx → σ≥My is in lw and there is the right commutative diagram above with σ≥Mx → x, σ≥My → y and σ≥M f : σ≥Mx → σ≥My are in u. Thus by the two out of three property of u, f is also in u. (ii) First we will show that Fwb C is a wst-thick subcategory of F C . Let x y z be a coﬁbration sequence x y y/x in F C . Then for each natural number n, since w is admissible, if two of in, in and in are in w, then the third one is also in w.

xn / / yn / / yn/xn x y y/x in in in xn+1 / / yn+1 / / yn+1/xn+1.

Namely if two of x, y and y/x are in Fwb C , then the third one is also in Fwb C .

9 Next let f : x → y be a morphism in wst and assume that x (resp. y) is in Fwb C . Then there exists a natural x y number N such that in (resp. in) and fn are in w for all n ≥ N. Then by considering the commutative diagram y x below and the two out of three property, it turns out that in (resp. in) is also in w.

fn xn / yn x y in in xn+1 / yn+1. fn+1

Thus y (resp. x) is also in Fwb C . Next let S be a wst-prethick subcategory of F C which contains j(C ). We will show that S contains x Fwb C . For an object x in Fwb C , there is an integer N such that in is in w for all n ≥ N. Then there is a zig-zag sequence j(xN) ← σ≥N j(xN) → x of morphisms in wst. Since S is wst-closed, it contains x.

(iii) If y is in Fb C , then by the same proof of 3.4 (ii), we can show that there exists a factorization (i f , p f ,u f ) of f with u f ∈ ObFb C and p f ∈ lw. x y In general case, let N be a natural number such that in = idxN and in ∈ w for any n ≥ N. Then there exists 0 0 f p 0 a factorization x → σ≤Nx → y of f with p ∈ wst. Now by applying the argument in the previous paragraph to f 0, we obtain the desired factorization.

j Corollary 3.9. The inclusion functor C = (C ,w) → (Fb C ,wst) → (Fwb C ,wst) induces a homotopy equivalence K(C) → K(Fwb C ;wst) of K-theory.

Proof. By approximation theorem in [Sch06, A.2] and 3.8 (iii), the inclusion functor (Fb C ,wst) ,→ (Fwb C ,wst) induces a homotopy equivalence of spectra on K-theory. Thus combining with 3.7 (ii), we obtain the result.

Definition 3.10. We denote the smallest admissible class of morphisms in F C which contains lw and ΘF C by wsst and call it the class of strongly stably weak equivalences in F C . We write wsus for the admissible class wFwb C of morphisms in F C which corresponds to the wst-prethick subcategory Fwb C of F C and call it the class of suspension weak equivalences in F C . We write F C and SC for the pairs (F C ,wsst) and (F C ,wsus) and call them flasque envelope of C and the suspension of C respectively. The naming of FC and SC are justified by the following results.

Proposition 3.11. Assume that C is essentially small. Then (i) (Eilenberg swindle). K(FC) is trivial. ∼ (ii) (Suspension). There exists a natural homotopy equivalence of spectra K(SC) → ΣK(C). F Proof. (i) We denote the functor F C → F C which sends an object x in F C to n≥0 x[n] by F. Then there exists an equality F = F[1] t idF C and there exists a natural weak equivalence θF : F → F[1] with respect to wsst. Thus K(F) = K(F[1]) on K(FC) and the identity morphism of K(FC) is trivial. Hence K(FC) is trivial. (ii) By ﬁbration theorem in [Sch06, A.3] and 3.9, there exists a ﬁbration sequence K(C) → K(FC) → K(SC) ∼ of spectra. Combining with the result (i), we obtain the homotopy equivalence of spectra K(SC) → ΣK(C).

Deﬁnition 3.12 (Non-connective K-theory for strictly derivable Waldhausen categories). For an essentially small strictly derivable Waldhausen category C = (C ,w), by Proposition 3.11 (ii), there exists a canonical morphism K(C) → ΩK(SC). (10) 2 We denote the spectrum K(C) to be the sequence of spaces K(C), K(SC), K(S C) with the structure morphisms given by (10).

10 References

[Cis10] Denis-Charles Cisinski, Invariance de la K-theorie´ par equivalences´ deriv´ ees´ , Jorunal of K-theory 6 (2010), p.505-546.

[Sch06] Marco Schlichting, Negative K-theory of derived categories, Math. Z. 253 (2006), p.97-134. [Wal85] Friedhelm Waldhausen, Algebraic K-theory of spaces, In Algebraic and geometric topology, Springer Lect. Notes Math. 1126 (1985), p.318-419.

SATOSHI MOCHIZUKI DEPARTMENT OF MATHEMATICS, CHUO UNIVERSITY, BUNKYO-KU, TOKYO, JAPAN. e-mail: [email protected]