LECTURE . EXACT ∞-CATEGORIES
We now introduce the higher categorical analogue of uillen’s theory of exact categories.
.. Definition. A triple of ∞-categories, or just a triple (C , C† , C † ) consists of an ∞-category C , a pair structure
(C , C† ) on C , and a pair structure (C , C † ) on C op . e morphisms of C† will be called ingressive morphisms or
cofibrations, whereas the morphisms of C † will be called egressive morphisms or fibrations.
We write Trip∞ := Pair∞ ×Cat∞ Pair∞ .
.. It is obvious from this definition that Trip∞ is compactly generated.
.. Definition. (..) Suppose (C , C† , C † ) a triple of ∞-categories. A pullback (respectively, pushout) square
X : Δ1 × Δ1 . C , denoted
X00
.
X01
.
.
X10
.
X11
.,
is said to be ambigressive if X10 . X11 is ingressive and X01 . X11 is egressive (resp., if X00 . X01 is
ingressive and X00 . X10 is egressive).
(..) An exact ∞-category (C , C† , C † ) is a triple of ∞-categories satisfying the following conditions.
(...) e pair (C , C† ) is a Waldhausen ∞-category.
(...) e pair (C , C † ) is a coWaldhausen ∞-category.
(...) A square in C is an ambigressive pullback if and only if it is an ambigressive pushout.
(..) If C and D are exact ∞-categories, then a functor C . D is exact just in case it is exact both as a functor
of Waldhausen ∞-categories and as a functor of coWaldhausen ∞-categories.
ough there may seem to be a chance of ambiguity here, in fact there is none.
.. Example. (..) e nerve NC of an ordinary category C can be endowed with a triple structure yielding
an exact ∞-category if and only if C is an ordinary exact category, in the sense of uillen, wherein the
admissible monomorphisms are exactly the cofibrations, and the admissible epimorphisms are exactly the
fibrations.
(..) At the other extreme, any stable ∞-category is an exact ∞-category in which all morphisms are both egressive and ingressive, and, conversely, any ∞-category that can be regarded as an exact category with the maximal triple structure (in which all maps are bigressive) is a stable ∞-category.
(..) omason–Trobaugh called a triple (C, C† , C† ) of ordinary categories satisfying the first two of the conditions above a category with bifibrations. When a class of weak equivalences is included, omason–Trobaugh
use the term biWaldhausen category. However, their notion does not require that ambigressive pushout
squares coincide with ambigressive pullback squares.
(..) For a less trivial example, suppose A a stable ∞-category equipped with a t-structure. In A≥0 , every pushout
square is a pullback square, but not conversely. Consequently, we can define an exact ∞-category structure
on A≥0 in which every morphism is ingressive, but the egressive morphisms are precisely those morphisms
Y . Z such that for any object W ∈ A ♡ , the map
[Z, W] .
[Y, W]
is injective. Dually, we can define an exact ∞-category structure on A≤0 in which every morphism is egressive, but the ingressive morphisms are precisely those morphisms X . Y such that for any object W ∈ A ♡ ,

the map
[W, X] .
[W, Y]
is injective.
.. Lemma. In an exact ∞-category, a pushout (respectively, pullback) square
X00
.
X01
.
.
X10
.
in which the morphism X00 .
(resp., pushout) square.
X11
.
X01 is ingressive (resp., in which the morphism X01 .
.. Lemma. In an exact ∞-category, a morphism X .
X11 is egressive) is also a pullback
Y is ingressive just in case there is a fiber sequence
X.
Y.
.
0.
in which Y .
Z is egressive. Dually, a morphism Y .
Z.
Z is egressive just in case there is a cofiber sequence
X.
Y.
.
0.
in which X .
Z.
Y is ingressive.
Consequently, one of these classes in an exact ∞-category determines the other.
.. Exercise. Convince yourself that for an exact ∞-category C , the “co-K-theory” — i.e., the universal functor on
coWaldhausen ∞-categories that splits fiber sequences, i.e., the K-theory of C op — agrees with the K-theory.
.. Notation. Write Exact∞ for the subcategory of Trip∞ consisting of exact ∞-categories and exact functors.
.. Lemma. e opposite inolution on Trip∞ restricts to an autoequivalence of Exact∞ .
.. Proposition. e ∞-category Exact∞ admits all small limits, and the inclusion functor Exact∞ .
preserves them.
Proof. e only nontrivial point is to note that, for any small ∞-category A and any functor A .
the assignment a . Ca , if C is the limit of the composite functor
A.
Exact∞ .
Trip∞
Exact∞ given by
Trip∞ ,
then a square is an ambigressive pushout if and only if it is an ambigressive pullback.
□
.. Proposition. e ∞-category Exact∞ admits all small filtered colimits, and the inclusion functor Exact∞ .
preserves them.
Trip∞
Proof. Again, the only nontrivial point is to show that, for any small filtered ∞-category A and any functor A .
given by the assignment a . Ca , if C is the colimit of the composite functor
Exact∞
A.
Exact∞ .
Trip∞ ,
then a square is an ambigressive pushout if and only if it is an ambigressive pullback. For this, suppose that Y′ . Y
a cofibration of C , and suppose Y′ . X′ a fibration of C . Without loss of generality, one may assume that there

exists an object a of A and a cofibration Y′a .
canonical functor of triples Ca . C are Y′ .
Ya of Ca and a fibration Y′a . X′a of Ca whose images under the
Y and Y′ . X′ , respectively. One may form the pushout square
Y.′a
Y.a
.
X.′a
X.a
in Ca , which is also a pullback square. e image under the canonical functor of triples Ca .
gressive pullback
Y.′
C is then an ambi-
Y.
.
X.′
X..
us any ambigressive pushout is also an ambigressive pullback. e same argument for the functor a .
that any ambigressive pullback is also an ambigressive pushout.
op
Ca shows
□
.. Proposition. e ∞-category Exact∞ admits finite direct sums.
Proof. is follows immediately from the analogous assertions for Wald∞ and coWald∞ .
... Corollary. e ∞-category Exact∞ admits all tiny coproducts.
In fact, we have the following.
.. Proposition. e ∞-category Exact∞ is compactly generated.

□