WALDHAUSEN’S S• -CONSTRUCTION

VIKTORIYA OZORNOVA AND MARTINA ROVELLI

arXiv:2412.17398v1 [math.AT] 23 Dec 2024

Abstract. This note is an expository contribution for a proceedings volume of

the workshop Higher Segal Spaces and their Applications to Algebraic K-theory,
Hall Algebras, and Combinatorics. We survey various versions of Waldhausen’s S• -
construction and the role they play in defining K-theory, and we discuss their 2-
Segality properties.

Introduction

A prominent invariant for a ring (spectrum) R is its K-theory space – which is in fact
a spectrum – K(R). Various constructions are available to access K(R), most notably:
Quillen’s +-construction [Qui72], Quillen’s Q-construction [Qui73], Segal’s group comple-
tion [Seg74], Grayson’s S −1 S-construction [Gra76], and Waldhausen’s S• -construction
[Wal85]. The latter allows for some of the most general kinds of input, and it is the focus
of this note.
From any ring R, one can consider the exact category ER of finitely generated projective
R-modules. Further, given E any exact (∞-)category – or even something more general
such as a Waldhausen (∞-)category – its K-theory space K(E) can be obtained through
the realization of a certain simplicial space: Waldhausen’s S• -construction S• (E).
A lot of the structure of E – such as information about the bicartesian squares in E – is
naturally retained in the simplicial space S• (E). Similarly, a lot of the relevant properties
of E – such as the fact that bicartesian squares are determined by the span of which they
are a colimit cone – are manifested in an interesting way in S• (E): they are encoded in the
fact that S• (E) is a (lower) 2-Segal space. Roughly, S• (E) is a 2-Segal space if E is pointed
and stable in an appropriate sense, and it is a lower 2-Segal space if E is pointed and
satisfies just one half of the stability requirements, which we may refer to as semi-stable.
The notion of a 2-Segal space was discovered independently in [DK19] (cf. the contribu-
tion [Ste24]) and in [GCKT18] under the name of decomposition space (cf. the contribution
[Hac24]), and the S• -construction S• E was identified as the main example by both sets
of authors. It was later shown in [BOO+ 18, BOO+ 21a] (cf. the contribution [Rov24])
that in fact the S• -construction defines an equivalence between 2-Segal spaces and cer-
tain double Segal spaces. Hence, every 2-Segal space does indeed arise as an appropriate
S• -construction.
In most cases of interest for E, the K-theory space K(E) can be delooped infinitely
many times, so it is in fact an Ω-spectrum. This can be done by realizing an appropriate
(n)
iterated version of the S• -construction, which produces a multisimplicial space S•,...,• (E).
Again, a lot of the structure of E is manifested as a 2-Segal property in each simplicial
variable individually, so that it should be a multisimplicial space that is (lower) 2-Segal
(n)
in each simplicial variable. Enhancing the previous picture, S•,...,• (E) should be an n-fold
2-Segal space if E satisfies properties of pointedness and stability, and it should be an
n-fold lower 2-Segal space if E satisfies only one half of the stability requirements.
1
 (n)
E S• (E) S•,...,• (E)
stable pointed 2-Segal n-fold 2-Segal
semi-stable pointed lower 2-Segal n-fold lower 2-Segal
In this note we discuss the various flavors of S• -constructions, how they can be used
to define K-theory, their 2-Segality properties, and how this is remembering features of
the original input.

Acknowledgements. We are grateful to Julie Bergner, Joachim Kock, Mark Penney,

and Maru Sarazola for organizing a wondeful workshop at the Banff International Re-
search Station: Higher Segal Spaces and their Applications to Algebraic K-theory, Hall
Algebras, and Combinatorics (24w5266, originally 20w5173). We are also thankful to
Claudia Scheimbauer for creating and letting us use the picture macros from our previ-
ous papers and to Lennart Meier for helpful discussions on the history of the subject.
MR acknowledges the support from the National Science Foundation under Grant No.
DMS-2203915.

1. Waldhausen’s S• -construction and K-theory space

We review Waldhausen’s S• -construction and some of its generalizations.

1.1. Waldhausen’s S• -construction. We discuss in detail what Waldhausen’s S• -construction

looks like and how it is used to define K-theory in the main example: an exact category
of modules over a ring.

1.1.1. The sequence construction. Let R be a nice (unital associative commutative) ring
and let ER be the category of modules over R or a full subcategory thereof.
The easiest case to keep in mind throughout this section is the case of R being any
ring and ER being the category of all R-modules, which is a primary example of an
abelian category. However, this choice leads to a trivial K-theory (cf. Remark 1.13).
Alternatively, one can consider the case of R being a local regular ring and ER being
the category of finitely generated R-modules, which is also an abelian category. Overall,
what is considered the most interesting case is that of R being any ring and ER being the
category of finitely generated projective R-modules, which fails to be an abelian category,
but can be given the structure of an exact category.
When drawing diagrams valued in ER , we follow the convention that arrows drawn
horizontally, resp. vertically, as

A0
A0 A1 resp.
A1

correspond to an injective (resp. surjective) morphism in ER , that displayed squares of

the form

A00 A10

A10 A11

2
are bicartesian squares, and that objects denoted Zi are R-trivial modules, i.e., modules
with one element.
Some first relevant information about the ring R which is naturally stored in ER is the
understanding of sequences of subobjects. For instance, if EZ is the category of finitely
generated projective abelian groups, there are objects which admit non-trivial chains of
subobjects of arbitrary length. Instead, if EZ/2 is the category of finitely generated Z/2-
modules, the length of a chain of subobjects of a given object is bounded by the rank of
this object. So one could consider:
Construction 1.1. For n ≥ 0, let Seqdisc
k (ER ) denote the set of all sequences of subobjects
in ER of length k. For instance, for k = 3 the generic element is of the form

A1 A2 A3

For 0 < i ≤ k, let the i-th face map di : Seqdisc

k ER → Seqdisc
k−1 (ER ) be given by skipping
the i-th term. For instance, if k = 3 this gives:


 A2 A3 , i = 1







A1 A2 A3 7−→d i A1 A3 , i = 2








 A1 A2 , i = 3

Let the 0-th face map d0 : Seqdisc disc

k (ER ) → Seqk−1 (ER ) be given by taking quotients modulo
the first term. For instance, if k = 3 this gives:

A1 A2 A3 d0 A2 /A1 A3 /A1
7−→

For 0 ≤ i ≤ k, let the i-th degeneracy map si : Seqk ER → Seqk+1 (ER ) be given by
doubling the i-th term for i > 0 and adding a zero object 0 for i = 0. For instance, if
k = 2 this gives


 0 A1 A2 , i = 0







si
A1 A2 7−→ A1 A1 A2 , i = 1








 A1 A2 A2 , i = 2

As discussed in [Wal78, §4], the construction Seqdisc

k (E) is however not natural in the
variable k:
Remark 1.2. The assignment [k] 7→ Seqdisc
k (ER ) does not define a simplicial set. Given σ
in Seqdisc
3 (E R )

A1 A2 A3

3
many simplicial identities hold, such as

d1 d2 (σ) = A3 = d1 d1 (σ).

However, any expression involving the face d0 is problematic. For instance, when compar-
ing d0 d0 to d0 d1 , the best that one can say is that there is an isomorphism - yet generally
not an equality - of the form

(1.3) d0 d0 (σ) = (A3 /A1 )/(A2 /A1 ) ∼

= A3 /A2 = d0 d1 (σ).

Overall, the construction Seqdisc

k (ER ) suffers from the following shortcomings:

(1) The assignment [k] 7→ Seqdisck (ER ) does not define a simplicial set of sequences in ER ,
as discussed in Remark 1.2.
(2) The definition of Seqdisc
k (ER ) breaks the natural symmetry present in the structure of
ER , in that it prioritizes subobjects over quotients.
(3) The set of sequences Seqdisc
k (ER ) is not so meaningful in itself, in that it is not capable
of recognizing when two sequences are isomorphic, and should instead be replaced
with the groupoid or space of sequences.

A way to address at least some of these issues is to consider the following variant (which
can be seen as a variant of [Wal85, §1] and as an instance of [Wal78, §4], taking in ER
cofibrations to be monomorphisms and weak equivalences to be isomorphisms):

Construction 1.4. For k ≥ 0, let Seqk (ER ) denote the groupoid of all sequences of
n subobjects in ER ; that is, the maximal groupoid in the category of functors from [k]
to ER spanned by the set of objects Seqdisc k (ER ). For 0 ≤ i ≤ k, let the i-th face map
di : Seqk (ER ) → Seqk−1 (ER ) and the i-th degeneracy map si : Seqk (ER ) → Seqk+1 (ER ) to
be defined by extending the formulas from Construction 1.1 in the obvious way.

By making appropriately coherent the isomorphism witnessing the failure of the sim-
plicial identities for Seqk (ER ) – e.g. the one from (1.3) – one could show:

Proposition 1.5. The assignment [k] 7→ Seqk (ER ) defines a pseudo simplicial groupoid.

The construction Seqk (ER ) addresses (3), but it only partially addresses (1) – in that
it involves a higher categorical metatheory – and it does not address (2).

1.1.2. The S• -construction. An attempt at solving the issues (1), (3) and (2) going in a
different direction is the following (cf. [Wal85, §1.4]):
Denote by Ar[k] = [k][1] the arrow category of [k]; that is, the category of functors
[1] → [k] and natural transformations.

Construction 1.6. For n ≥ 0, let Skdisc (ER ) denote the set of exact functors Ar[k] → ER .
That is, of functors Ar[k] → ER which:

• send the objects of the form (i, i) to a zero object in ER ,

• send the morphisms of the form (i, j) → (i, j + ℓ) to monomorphisms in ER ,
• send the morphisms (i, j) → (i + k, j) to epimorphisms in ER ,
• sends the commutative squares involving two maps of the form (i, j) → (i + k, j) and
two maps of the form (i, j) → (i, j + ℓ) to bicartesian squares in ER .

For instance, a functor Ar[3] → ER defining an object of S3disc (ER ) is of the form
4
 Z0 A01 A02 A03

Z1 A12 A13

Z2 A23

Z3

with the convention that all objects featuring on the diagonal are zero objects, all hori-
zontal morphisms are monomorphisms, all vertical morphisms are epimorphisms, and all
squares are bicartesian squares. For 0 ≤ i ≤ k, let the i-th face map di : Skdisc (ER ) →
disc
Sk−1 (ER ) be given by skipping the i-th row and the i-th column. For instance, if k = 3
the face maps d2 and d3 act as:

Z0 A01 A02 A03 Z0 A01 A03

Z1 A12 A13 Z1 A13

d2
7−
−→
Z2 A23

Z3 Z3

Z0 A01 A02 A03 Z0 A01 A02

Z1 A12 A13 Z1 A12

d3
7−
−→
Z2 A23 Z2

Z3

For 0 ≤ i ≤ k, let the i-th degeneracy map si : Skdisc (ER ) → Sk+1

disc
(ER ) be given by by
doubling the i-th term. For instance, if k = 2 the degeneracy maps s0 and s1 are as
follows

Z0 Z0 A01 A02
Z0 A01 A02
Z0 A01 A02
Z1 A12 −s0
7−→
Z1 A12
Z2
Z2

5
 Z0 A01 A01 A02
Z0 A01 A02
Z1 Z1 A12
Z1 A12 −s1
7−→
Z1 A12
Z2
Z2

One can then check that the construction Skdisc (E) is natural in k:

Proposition 1.7. The assignment [k] 7→ Skdisc (ER ) defines a simplicial set S•disc (ER ).

The construction Skdisc (ER ) solves (1) without requiring higher categorical metatheory,
and it solves (2), but does not fix (3). So one can then mix the two adjustments Seqk (ER )
and Skdisc (ER ), culminating with Waldhausen’s original S• -construction (cf. [Wal85, §1.3]
in the special case that weak equivalences are chosen to be isomorphisms):

Construction 1.8. For k ≥ 0, let Sk (ER ) denote the groupoid of exact functors from
Ar[k] to ER ; that is, the maximal groupoid in the category of functors Ar[k] to ER spanned
by the set of objects Skdisc (ER ). For 0 ≤ i ≤ k, let the i-th face map di : Sk (ER ) →
Sk−1 (ER ) and the i-th degeneracy map si : Sk (ER ) → Sk+1 (ER ) be defined by extending
the formulas from Construction 1.6 in the obvious way.

One can then check that this produces a simplicial object (cf. [Wal85, §1.3]), which we
refer to as the S• -construction of ER :

Proposition 1.9. The assignment [k] 7→ Sk (ER ) defines a simplicial groupoid S• (ER ).

The following proposition explains how the construction S• (ER ) can really be under-
stood as an appropriate correction of Seq• (ER ), in that (cf. [DK19, Lemma 2.4.9]):

Proposition 1.10. For k ≥ 0, the canonical projection on “the 0-th row” is an equiva-
lence of groupoids
≃
Sk (ER ) −
→ Seqk (ER ).

Idea. For instance, for k = 3 this map of groupoids can be depicted as

Z0 A01 A02 A03 A01 A02 A03

Z1 A12 A13
7→
Z2 A23

Z3

By completing including the zero module in a few positions of the diagram and completing
all the relevant spans to pushout squares in ER , which can be done is an essentially unique
way, one can consider the assignment
6
 A01 A02 A03 0 A01 A02 A03

0 P12 P13
7→
0 P23

and verify it defines an inverse equivalence of groupoids

≃
Seqk (ER ) −
→ Sk (ER ).
for the desired map. The statement for general k can be proven through an appropriate
enhancement of this argument. 

We can take a closer look at how, besides for the sequences from Proposition 1.10,
many other relevant “homological” features of R are retained in S• (ER ):
Remark 1.11. We have that:
(0) If Z(ER ) denotes the groupoid of zero objects in ER , there is an isomorphism of
groupoids
S0 (ER ) ∼
= Z(ER ).
(1) If Ob(ER ) denotes the core of ER , i.e., the maximal groupoid inside ER , as a special
case of Proposition 1.10 there is an equivalence of groupoids
S1 (ER ) ≃ Ob(ER ).
(2) If ExSeq(ER ) denotes the groupoid of short exact sequences in ER , there is an equiv-
alence of groupoids
S2 (ER ) ≃ ExSeq(ER ),
which can be depicted as

Z0 A01 A02 A01 A02

Z1 A12 7→ A12

Z22

(3) If BiCartSq(ER ) denotes the groupoid of bicartesian squares involving two monomor-
phisms and two epimorphisms in ER , there is an equivalence of groupoids
S3 (ER ) ≃ BiCartSq(ER ),
which can be depicted as

A00 A01 A02 A03 A02 A03

Z1 A12 A13 A12 A13

7→
Z2 A23

Z3

with inverse equivalence given by

7
 A02 A03 0 P01 A02 A03

A12 A13 0 A12 A13

7→
0 P23

From S• (ER ), one can define the K-theory space K(ER ) (cf. [Wal85, §1.3]), by further
op op
applying Real : Gpd ∆ → S ∆ → S, the realization of simplicial groupoids through
simplicial spaces:
Construction 1.12. The K-theory space of ER is
K(ER ) := Ω(Real(S• (ER ))).
Remark 1.13. If ER is the category of modules over R, then K(ER ) is contractible (see
e.g. [Wei13, §II.2]). Instead, if ER is the category of finitely generated projective modules
over R, then K(ER ) is typically interesting, and it is what one refers to as the K-theory
space K(R) of the ring R. The homotopy groups of K(R) have been fully computed for
the case of R being a finite field (see [Qui72, Theorem 8]), and are only partially known
for R = Z (see [Wei05]).

We will see later in Section 2 how the space K(ER ) can be delooped infinitely many
times, so it actually comes from an Ω-spectrum.

1.2. Generalized S• -construction. We now mention the generalizations of Waldhausen’s

construction that have been studied in the literature, culminating with a detailed descrip-
tion in Section 1.2.3 for a rather general one.

1.2.1. Pointed stable contexts. We collect here all extensions of Waldhausen’s S• -construction
to a generalized input E that maintains a version of the properties of “stability” and
“pointedness” (or more generally “augmentation”). The structure of E needs to include
a class of objects, a class of distinguished objects, two classes of morphisms – horizontal
and vertical – and a class of squares. The properties required of E roughly encode the
following, to be understood in a strict or weak sense, depending on the context:
⋄ pasting: the fact that the horizontal morphisms can be composed, vertical morphisms
can be composed, and squares can be composed both horizontally and vertically;
⋄ pointedness (or more generally augmentation): the fact that there is a distinguished
class of objects – typically the class of zero objects of some sort – with the property
that every object receives uniquely a horizontal morphism from an object in this dis-
tinguished class, and every object maps uniquely through a vertical morphism to an
object in this distinguished class;
⋄ stability: the fact that every square is uniquely determined by the span in its boundary
and also uniquely determined by the cospan in its boundary, through the depicted
assignments:

A02 A03 A02 A03 A03

(1.14) 7→ 7→
A12 A12 A13 A12 A13

8
 Categorical structures that follow this scheme, roughly in the chronological order in
which they were considered, include the case of E being:

• an abelian category; a motivating example would be the abelian category of abelian

groups, or more generally the abelian category of modules over a ring R.
• a (proto)exact category, as in [Qui73, §1.2] and [DK19, §2.4]; a motivating example
would be the exact category of finitely generated abelian groups, or more generally the
exact category of finitely generated modules over a ring R, or the proto-exact category
of finite pointed sets;
• a stable ∞-category, as in [Lur06]; a motivating example would be the stable ∞-category
of spectra, or more generally the stable ∞-category of (compact) modules over a ring
spectrum.
• an exact ∞-category, as in [DK19, §7.2] (cf. also [Bar15]); a motivating example would
be the exact ∞-category of connective spectra.
• a stable pointed double category (or more generally a stable augmented double category),
as in [BOO+ 18, §3]; a motivating example is the path construction of a (reduced) 2-
Segal set (and in fact it is shown as [BOO+ 18, Theorem 6.1] that all examples are of
this kind).
• a stable pointed double Segal space and more generally a stable augmented augmented
double Segal space, as in [BOO+ 21a, §2.5]; a motivating example is the path construc-
tion of a (reduced) 2-Segal space (and in fact it is shown as [BOO+ 21a, Theorem 6.1]
that all examples are of this kind).
• a CGW category as in [CZ22, §2], (cf. also an ACGW category as in [CZ22, §5], and an
ECGW category as in [SS21, Part 1]); a motivating example would be the category of
reduced schemes of finite type.
• an isostable squares category as in [CS24, §3]; a motivating example is the squares
category of polytopes from [CS24, Examples 2.14]

These different frameworks can be organized as follows:

abelian categories stable ∞-categories

exact categories exact ∞-categories

CGW categories
stable pointed stable pointed
double categories isostable squares ? double Segal spaces
categories
stable augmented stable augmented
double categories double Segal spaces

Here, the horizontal direction roughly encodes the generalization from the strict to a
weaker metatheory, while the vertical direction progressively favors structure over prop-
erties.
Some existing comparisons between these frameworks include: [Qui73, §2] for how
exact categories recover abelian categories, [DK19, Example 7.2.3] for how exact ∞-
categories recover exact categories, [Nee90] for how exact ∞-categories possibly recover
exact categories through the derived ∞-category of chain complexes, [Lur18, §1.3.2] for
how stable ∞-categories recover abelian categories, [CZ22, Example 3.1] for how CGW
categories recover exact categories, [CS24, Proposition 3.11] for how isostable squares
9
categories recover stable pointed double categories, [BOO+ 21b] for how stable augmented
double Segal spaces recover exact categories and stable ∞-categories through appropriate
nerve constructions.
The study of how CGW categories recover pointed stable double categories and of
how pointed stable double Segal spaces recover CGW categories is subject of ongoing
work by some workshop participants (cf. [BLLM22, pp. 29-30]) A comment on how CGW
categories possibly relate to isostable square categories is in [CS24, Remark 3.27].
An appropriate S• -construction was considered for all these situations. This was done
in [Wal85, §1.3] for exact categories (as a special case of Waldhausen categories, there
referred to as categories with cofibrations and weak equivalences), in [Bar15, Recollec-
tion 5.8] (as a special case of Waldhausen ∞-categories) and [DK19, §2.4] for exact
∞-categories, in [BOO+ 18, §3] for pointed or augmented stable double categories, in
[BOO+ 21a, §2, §7] for pointed or augmented stable double Segal spaces, and in [CS24,
§2.2] for isostable squares categories. Many are recovered as a special case of the S• -
construction for Σ-spaces, which will be recalled in Section 1.2.3.
Results establishing the compatibility between the various S• -constructions include
[BOO+ 21b, §2, §3] for the compatibility of the one for exact categories and stable aug-
mented double categories with the one for stable augmented double Segal spaces, and
[CS24, Proposition 3.6] for the compatibility of the one for stable augmented double
categories with the one for isostable squares categories.
The versions of S• -constructions considered for different frameworks, however, don’t
always agree in a strict sense. For instance, the S• -construction for abelian categories and
the one for stable ∞-categories only agree up to equivalence upon taking the loop space
of its geometric realization, essentially as an instance of the Gillet–Waldhausen theorem
(see [TT90, Theorem 1.11.7]).

1.2.2. Pointed semi-stable context. One could also generalize in a different direction, drop-
ping one half of the stability axioms, asking for a square to be determined by the span
contained in its boundary; that is, only the left half of (1.14). For instance, this is the case
when the class of distinguished squares are certain pushout squares in a given category,
which are not necessarily pullbacks. Categorical structures fit this framework, roughly in
the chronological order in which they were considered, include the case of W being:

• a Waldhausen category as in [Wei13, Definition II.9.1.1] (originally referred to as cat-

egory with cofibrations and weak equivalences in [Wal85, §1.2]); a motivating example
would be the Waldhausen category of pointed CW-complexes, or more generally the
category of retractive spaces over a CW-complex X (see [Wal85, §1.1];
• a Waldhausen ∞-category, as in [Bar16, Definition 2.7]; this notion offers a common
framework to talk about stable ∞-categories and Waldhausen categories;
• a proto-Waldhausen squares category, as in [CS24, §2.3];
• a semi-stable pointed/augmented double category, by which we mean a double category
that satisfies only one half of the stability property from [BOO+ 18, §3]; that is, the one
corresponding to the left side of (1.14).
• a semi-stable pointed/augmented double Segal space, by which we mean a double Segal
space that only one half of the stability property from [BOO+ 21a, §2.5]; that is, the
one corresponding to the left side of (1.14).

These different frameworks could be organized as follows:

10
 Waldhausen
Waldhausen categories
∞-categories

semi-stable pointed ? protoWaldhausen ? semi-stable pointed

double categories squares categories double Segal spaces

semi-stable augmented semi-stable augmented

double categories double Segal spaces

An appropriate S• -construction was considered for several of these situations. This

was done in [Wal85, §1.3] for W being a Waldhausen category, in [Bar16, §5] (through
a slightly different though equivalent formalism) for W being a Waldhausen ∞-category,
and as [CS24, §2.2] for W being a proto-Waldhausen squares category. An S• -construction
for stable augmented double Segal spaces (in fact for any Σ-space) is given in Section 1.2.3.
Sometimes, however, the versions of S• -construction considered for different framework
don’t agree in a strict sense. For instance, as explained in [CS24, Remark 2.34], the S• -
construction for proto-Waldhausen categories and the one for Waldhausen categories only
agree up to equivalence upon taking geometric realization.

1.2.3. General context. We describe here a general framework for S• -construction recov-
ering many of the discussed instances.
Let Σ denote the category from [BOO+ 21a, Definition 2.9], obtained by freely adding a
op
terminal object [−1] to the category ∆ × ∆, and consider the category S Σ of Σ-spaces.
Roughly speaking, a Σ-space X consists of a bisimplicial space X•,• together with an
augmentation space X−1 and an augmentation map X−1 → X0,0 .
We briefly recall how the framework of Σ-spaces recovers the framework of exact cat-
egories through an appropriate nerve construction, as done in [BOO+ 21b, §2]. Studied
nerve constructions for other frameworks from Sections 1.2.1 and 1.2.2 include [BOO+ 21b,
§3] for stable ∞-categories and [BOO+ 21b, §4] for exact ∞-categories.
The datum of an exact category E consists of a category endowed with two classes
of distinguished morphisms, called admissible monomorphisms and admissible epimor-
phisms, and a distinguished class of objects. We refer the reader to [Qui73, §2] for a
complete definition, and we just mention that the axioms imposed on E guarantee that
zero objects exist, that admissible monomorphisms (resp. admissible epimorphisms) can
be composed, that pullback (resp. pushout) of an admissible monomorphism (resp. an
admissible epimorphism) exists and is still an admissible monomorphism (resp. an ad-
missible epimorphism). This is in particular enough structure to talk about admissible
short exact sequences and bicartesian squares. As a consequence of the axioms, appropri-
ate versions of the properties of pointedness and stability follow. An exact functor is a
functor that preserves the admissible short exact sequences in an appropriate way.
In most examples of exact categories ER based on categories of modules over a ring
R, the admissible monomorphisms (resp. admissible epimorphisms) are the module maps
that are injective (resp. surjective), and the zero objects are the trivial modules.
If S denotes the category of spaces and ExCat denotes the category of exact categories
op
and exact functors, we recall the exact nerve construction N ex : ExCat → S Σ from
11
[BOO+ 21b, Definition 2.2]. Given an exact category E, let
ex
N−1 (E) := Z(E)
denote the groupoid of zero objects in E, and for a, b ≥ 0 we let
ex
Na,b (E) := Homex ([a] × [b], E)
denote the groupoid of exact functors from [a] × [b] to E; that is, of functors [a] × [b] → ER
which:
• send the morphisms of the form (i, j) → (i, j + ℓ) to monomorphisms in ER ,
• send the morphisms (i, j) → (i + k, j) to epimorphisms in ER ,
• sends the commutative squares involving two maps of the form (i, j) → (i + k, j) and
two maps of the form (i, j) → (i, j + ℓ) to bicartesian squares in ER .
ex
For instance, a functor [2] × [3] → ER defining an object of N2,3 (E) is of the form

A00 A01 A02 A03

A10 A11 A12 A13

A20 A21 A22 A23

with the convention that all horizontal morphisms are monomorphisms, all vertical mor-
phisms are epimorphisms, and all squares are bicartesian squares.
We now turn to the S• -construction in the context of Σ-spaces. Let p : Σ → ∆ denote
the ordinal sum given by [−1] 7→ [0] and [a, b] 7→ [a + 1 + b], as considered in [BOO+ 21a,
Definition 2.15] (see [Rov24] in this volume). If S denotes the category of spaces, there
is an induced adjunction
op op
P = p∗ : S ∆ ⇆ S Σ : p ∗ = S•
op op
between the category S ∆ of simplicial spaces and the category S Σ of Σ-spaces.
So we obtain the description of the S• -construction:
Construction 1.15. For k ≥ 0, and X a Σ-space, we let
Sk (X) := Map(P∆[k], X)
denote the space of maps of Σ-spaces P∆[k] → X. For 0 ≤ i ≤ n, the i-th face map
di : Sk (X) → Sk−1 (X) and the i-th degeneracy map si : Sk (X) → Sk+1 (X) are induced
by the cosimplicial structure of P∆[•], and can be described in a similar fashion to those
of Construction 1.6.

In total we get the S• -construction of any Σ-space X:

Proposition 1.16. If X is a Σ-space, the assignment [k] 7→ Sk (X) defines a simplicial
space S• (X).

The S• -construction for exact categories and the one for Σ-spaces are compatible
op
through the exact nerve N ex : ExCat → S Σ :
Remark 1.17. It is shown as [BOO+ 21b, Theorem 2.18] that there is a commutative
diagram of categories
S• op
ExCat Gpd ∆
N ex N∗
op op
SΣ S•
S∆
12
Indeed, for n ≥ 0 and E an exact category, there is a natural bijection
= N (Homex (Ar[k], E)) ∼
= Map(P∆[k], N ex E) ∼
Sk (N ex E) ∼ = N (Sk (E)) = (N∗ S)k (E).

Through this generalized version of the S• -construction, one can now define the K-
theory space for a very general type of input. If X is a Σ-space
K(X) := Ω(RealS• (X)).

1.3. 2-Segality properties. Recall the definition of 2-Segal space from

[DK19] (see [Ste24] in this volume) – a.k.a. decomposition space in [GCKT18] (see [Hac24]
in this volume) – and its generalization lower 2-Segal space from [Pog17, Definition 2.2].
We discuss how the properties of the input E reflect into properties its 2-Segality
op
S• -construction. We will compose any existing S• -construction valued in Gpd ∆ from
op op
previous sections with the levelwise nerve functor N∗ : Gpd ∆ → S ∆ , keeping the same
notation. The overall expectation is the following:

stable augmented S•
2-Segal spaces
double Segal spaces

semi-stable augmented S•
lower 2-Segal spaces
double Segal spaces ?

S•
Σ-spaces simplicial spaces

1.3.1. The 2-Segal case. We start by treating the stable pointed case, namely the case of
E being one of the structures from Section 1.2.1.
Theorem 1.18. If E is any of the structures discussed in Section 1.2.1, the S• -construction
S• E is a 2-Segal space.

This is shown as [GCKT18, Theorem 10.10] for E being an abelian category, as [DK19,
§2.4] for E being a (proto-)exact category, as [BOO+ 18, Theorem 4.8] for E being a stable
augmented double category, as [BOO+ 21a, Theorem 5.1] for E being a stable augmented
double Segal space, as [CS24, Theorem B] for E being an isostable squares category. We
briefly illustrate how to treat one of the lowest dimensional 2-Segal maps:

Idea. The 2-Segal map

(d2 , d0 ) : S3 E → S2 E ×S1 E S2 E
can be depicted as

A00 A01 A02 A03 A00 A01 A03

Z1 A12 A13 Z1 A13 A11 A12 A13

7→ ,
Z2 A23 Z2 A23

Z3 Z3 A33

13
Using the structure and properties of E, one can consider the assignment

A00 A01 A03 A00 A01 P A03

Z1 A12 A13
Z1 A13 Z1 A12 A13
, Z2 A23 7→
Z2 A23
Z3
Z3 Z3

and verify that it defines an inverse equivalence of groupoids

≃
S2 E × S 1 E S2 E −
→ S3 E.
The other map
(d1 , d3 ) : S3 (E) → S2 (E) ×S1 (E) S2 (E)
can be treated analogously. 

Remark 1.19. One could also see that S• (E) is a 2-Segal space by observing that its
edgewise subdivision esd(S• (E)) is a Segal space (which is closely related to Quillen’s
Q-construction Q(E) from [Qui73, §I.2]), hence by [BOO+ 20, Theorem 2.11] the S• -
construction S• (E) is a 2-Segal space.

It was shown in [BOO+ 21a] (and [BOO+ 18] for the discrete version of the statement,
see also [Rov24] in this volume) that in fact all 2-Segal spaces can be obtained through
the S• -construction:
Theorem 1.20. The S• -construction is an equivalence between 2-Segal spaces and stable
augmented double Segal spaces.

1.3.2. The lower 2-Segal case. We now treat the pointed semi-stable cases, namely the
case of W being one of those from Section 1.2.2. While one half of the proof of Theo-
rem 1.18 (and of the argument we sketched in Section 1.2.2) can be reproduced at this
further level of generality, the other half cannot.
In fact, lacking one half of the stability conditions, the S• -construction from Sec-
tion 1.2.2 does not generally define a 2-Segal space. To see this, let’s consider the following
explicit example (obtained by adjusting the one from [BOO+ 20, Example 4.3].
Remark 1.21. Let W denote the Waldhausen category of finite pointed CW-complexes
and pointed cellular maps, with cellular embeddings as cofibrations and homotopy equiv-
alences as weak equivalences (a special case of [Wal85, §1.1-1.2] with X being a point).
Then we can see that the 2-Segal map
(d2 , d0 ) : S3 (W) → S2 (W) ×S1 (W) S2 (W)
is not injective on connected components, hence in particular it is not an equivalence.
Indeed, if P is a finite 2-dimensional CW-complex which is not contractible but whose
suspension is contractible (for example, the classifying space of the perfect group from
[Hat02, Example 2.38]), the following two objects of S3 (W) live in different connected
components but map to the same connected component in S2 (W) ×S1 (W) S2 (W) under
the map (d2 , d0 ):
14
 ∗ P P CP ∗ P CP CP

∗ ∗ ΣP and ∗ ΣP ΣP

∗ ΣP ∗ ∗

∗ ∗

In particular, the simplicial space S• (W) is not a 2-Segal space.

Others are expected to suffer from the same flaw. However, one half of the proof of
Theorem 1.18 should survive:
Expectation 1.22. If W is any of the structures discussed in Section 1.2.2 the S• -
construction S• W is a lower 2-Segal space.

This was shown in [Car24, §7, §8] for W being a Waldhausen category, and should be
true for all listed examples.
One could study whether all lower 2-Segal spaces arise as S• -construction:
Question 1.23. Is this S• -construction an equivalence between lower 2-Segal spaces and
semi-stable augmented double Segal spaces? If not, what is its homotopy essential image?

This question is possibly treatable by adjusting the argument from [BOO+ 21a] (or
[BOO+ 18] for the discrete case).

2. Iterated Waldhausen’s S• -construction and K-theory spectrum

We discuss how to deloop K-theory through an appropriate iteration of Waldhausen’s

construction under appropriate assumptions.

2.1. Waldhausen’s iterated S• -construction. We recall Waldhausen’s technique to

deloop the K-theory space of ER .
Remark 2.1. If E is an exact category, there is an exact category [Ar[k], E]. The objects
are the exact functors from Ar[k] to E, and the exact structure is inherited from that of
E (and is essentially a special case of [Pen17, Lemma 3.15]). More precisely, the ordinary
S• -construction can be lifted to a functor
op
Se• := [Ar[•], −] : ExCat → ExCat ∆ .

Using this, and inspired by [Pen17, (3.6)] for the case n = 2, we can define the iterated
(n)
S• -construction – the multisimplicial groupoid S•,...,• (ER ) – as follows:
(n)
Construction 2.2. For n ≥ 0 and k1 , . . . , kn ≥ 0, let Sk1 ,...,kn (ER ) denote the groupoid
(n)
Sk1 ,...,kn (ER ) := Homex (Ar[k1 ] × · · · × Ar[kn ], ER )
of exact functors from Ar[k1 ] × · · · × Ar[kn ] to ER . That is, of functors from Ar[k1 ] × · · · ×
Ar[kn ] to ER which:
• send the objects of the form ((i1 , i1 ), . . . , (in , in )) to a zero object in ER ,
• send the morphisms of the form
((i1 , j1 ), . . . , (in , jn )) → ((i1 , j1 + ℓ1 ), . . . , (in , jn + ℓn ))
to monomorphisms in ER ,
15
• send the morphisms of the form
((i1 , j1 ), . . . , (in , jn )) → ((i1 + k1 , j1 ), . . . , (in + kn , jn ))
to epimorphisms in ER ,
• sends the commutative squares involving two maps of the form
((i1 , j1 ), . . . , (in , jn )) → ((i1 + k1 , j1 ), . . . , (in + kn , jn ))
and two maps of the form
((i1 , j1 ), . . . , (in , jn )) → ((i1 , j1 + ℓ1 ), . . . , (in , jn + ℓn ))
to bicartesian squares in ER .

So in total we get a multisimplicial groupoid:

(n)
Proposition 2.3. The assignment [k1 , . . . , kn ] 7→ Sk1 ,...,kn (ER ) defines an n-fold simpli-
(n)
cial groupoid S•,...,• (ER ).
(n)
One can understand S•,...,• as an analog n-fold iteration of S• , which justifies the
name “iterated” S• -construction (reconciling with the original viewpoint from [Wal85,
§1.3], [Wei13, §IV.8.5.5] and [Pen17, Corollary 3.18] for n = 2). Let’s explain this more
in detail for the case n = 2.
Remark 2.4. If Ob : ExCat → Gpd denotes the core functor, there is a commutative dia-
gram of categories:
e•
S op e• )∗
(S op op
ExCat ExCat ∆ (ExCat ∆ )∆
(((Ob)∗ )∗
(2)
S•,•
op op
((Gpd ∆ )∆
Indeed, given an exact category E and a, b ≥ 0, there are natural bijections for a, b ≥ 0:
Ob((Sea )∗ (Seb (E))) ∼
= Ob((Sea )∗ ([Ar[b], E]))
∼
= Ob([Ar[a], [Ar[b], E]])
∼
= Homex (Ar[a], [Ar[b], E])
∼ (2)
= Homex (Ar[a] × Ar[b], E) = Sa,b (E).
op op op op
Denote by Real(n) : Gpd ∆ ×···×∆ → S ∆ ×···×∆ → S the realization of n-fold sim-
plicial groupoids through n-fold simplicial spaces. It is a special case of [Wal85, §1.3]
(cf. [Wei13, §IV.8.5.5]) that the K-theory space K(ER ) can be delooped infinitely many
times as follows:
Theorem 2.5. For n ≥ 1, there is an equivalence of spaces
(n)
K(ER ) ≃ Ωn Real(n) (S•,...,• (ER )).

2.2. Generalized iterated S• -construction. In this last section we provide a more

general version of the iterated construction that would work for any Σ-space. This might
perhaps allow for an enhancement of Waldhausen’s proof of Theorem 2.5, delooping the
S• -construction in more general situations.
Let p(n) = (p, . . . , p) : Σ → ∆ × · · · × ∆ be given by σ 7→ (p(σ), . . . , p(σ)). There is an
adjunction
op op op (n) (n)
P (n) = p(n),∗ : S ∆ ×···×∆ ⇆ S Σ : p∗ = S•,...,•
op op op
between the category S ∆ ×···×∆ of n-fold simplicial spaces and the category S Σ of
Σ-spaces.
16
 We can understand the functor P (n) on the representable functor ∆[k1 , . . . kn ]:
Proposition 2.6. For n ≥ 1 and k1 , . . . , kn ≥ 0, there is an isomorphism of Σ-spaces:
∼ P∆[k1 ] × · · · × P∆[kn ] ∼
P (n) ∆[k1 , . . . kn ] = = N ex Ar[k1 ] × . . . N ex Ar[kn ].

Proof. Let’s do the case n = 2, the general case working similarly. For α in Σ, and
k1 , k2 ≥ 0, there are natural isomorphisms of spaces
∼ Map(Σ[α], p(2),∗ ∆[k1 , k2 ])
(P (2) ∆[k1 , k2 ])α =
∼ (2)
= Map(p! Σ[α], ∆[k1 , k2 ])
∼
= Map(∆[p(α), p(α)], ∆[k1 , k2 ])
∼
= Map(∆[p(α)], ∆[k1 ]) × Map(∆[p(α)], ∆[k2 ])
∼
= Map(Σ[α], P∆[k1 ]) × Map(Σ[α], P∆[k2 ])
∼
= (P∆[k1 ])α × (P∆[k2 ])α ,
as desired. 
(n) op op
×···×∆op
We can then describe the functor S•,...,• : S Σ → S∆ explicitly:
Construction 2.7. Given a Σ-space X, n ≥ 0 and k1 , . . . , kn ≥ 0, we set
(n)
Sk1 ,...,kn (X) = Map(P∆[k1 ] × · · · × P∆[kn ], X)
to be the space of maps of Σ-spaces P∆[k1 ] × · · · × P∆[kn ] → X.

So in total we get a multisimplicial space:

(n)
Proposition 2.8. Given a Σ-space X, the assignment [k1 , . . . , kn ] 7→ Sk1 ,...,kn (X) defines
(n)
an n-fold simplicial space S•,...,• (X).

This construction recovers the one for exact categories through the exact nerve N ex : ExCat →
Σop
S from [BOO+ 21b, Definition 2.2] as follows:
Remark 2.9. There is a commutative diagram of categories
(2)
S•,• op
×∆op
ExCat Gpd ∆
N ex N∗

Σop ∆op ×∆op

S (2)
S
S•,•

Indeed, for k1 , k2 ≥ 0 and E an exact category, there is a natural bijection

(2) ∼
Sk1 ,k2 (N ex E) = Map(P∆[k1 ] × P∆[k2 ], N ex E)
∼
= N Homex (Ar[k1 ] × Ar[k2 ], E)
∼ (2)
= N Sk1 ,k2 (E) ∼
= (N∗ S (2) )k1 ,k2 (E).
(n)
Remark 2.10. It is possible to recognize the functor S•,...,• as an appropriate iteration of
the S• -construction in the context of Σ-spaces, by replacing the category ExCat with the
op
category S ∆ in Remark 2.4.

One could then hope for a delooping in larger generality, so that K(X) would come
from an Ω-spectrum:
Question 2.11. For n ≥ 1, under which conditions on a Σ-space X does one have
equivalence of spaces
(n)
K(X) ≃ Ωn Real(n) (S•,...,• (X))?
17
 This type of statement was proven in [Wal85, §1.3] in the context of Waldhausen
categories, and in [SS21, §7] in the context of ECGW categories, and it might hold in
larger generality.

2.3. Multi-2-Segality properties. One could explore the 2-Segality property of the
iterated Waldhausen construction. Not much has been done in this direction, and we just
include some speculative considerations. The overall expectation is:
(n)
stable augmented S•,...,•
n-fold 2-Segal spaces
double Segal spaces ?

(n)
semi-stable augmented S•,...,•
n-fold lower 2-Segal spaces
double Segal spaces ?

(n)
S•,...,•
Σ-spaces n-fold simplicial spaces

2.3.1. The multi-2-Segal case. In the pointed stable case we should have (case n = 2 and
E being a protoexact category is [Pen17, Corollary 3.17]):
Expectation 2.12. If E is any of the structures discussed in Section 1.2.1, the iterated
(n)
S• -construction S•,...,• (E) is a 2-Segal object in each simplicial variable.

For instance, assuming that X P∆[ℓ] is a stable augmented double Segal space whenever
(2)
X is, Theorem 1.18 would imply that S•,ℓ (X) = S• (X P∆[ℓ]) is a 2-Segal space.
One could then study whether all multi-2-Segal spaces arise as an instance of an S• -
construction:
Question 2.13. Does the S• -construction define an equivalence between multi-2-Segal
spaces and stable augmented double Segal spaces? If not, is it injective (up to homotopy)?
And what is its (homotopy essential) image?

This question is possibly treatable by adjusting the argument from [BOO+ 21a] (or
[BOO+ 18] for the discrete case).

2.3.2. The multi-lower-2-Segal case. If W is any structure from Section 1.2.2, the S• -
construction is not expected to define a multi-2-Segal space, as flaws similar to those
highlighted in Remark 1.21 would remain. However:
Expectation 2.14. If W is any of the structures discussed in Section 1.2.2 the iterated
(n)
S• -construction S•,...,• (W) is a lower 2-Segal object in each simplicial variable.

For instance, assuming that X P∆[ℓ] is a semi-stable augmented double Segal space
(2)
whenever X is, Expectation 1.22 would imply that S•,ℓ (X) = S• (X P∆[ℓ]) is a lower
2-Segal space.
One could study whether all multi-lower-2-Segal spaces arise as S• -construction:
Question 2.15. Does the S• -construction define an equivalence between multi-lower-2-
Segal spaces and semi-stable augmented double Segal spaces? If not, is it injective (up to
homotopy)? And what is its (homotopy essential) image?

Again, this question is possibly treatable by adjusting the argument from [BOO+ 21a]
(or [BOO+ 18] for the discrete case).
18
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Max Planck Institute for Mathematics, Bonn, Germany

Email address: viktoriya.ozornova@mpim-bonn.mpg.de

Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst,

USA
Email address: mrovelli@umass.edu

20