Posted by Jason Polak on 26. December 2015 · Write a comment · Categories: math

Waldhausen categories are a type of categores that include abelian categories, exact categories, and some topological categories. They can be used to define the $K$-theory of these categories, and are more general than the exact category framework of Quillen as they allow categories like cell complexes, which was the original motivation of Friedhelm Waldhausen when he introduced them. This series of posts will attempt to explain the first part of Waldhausen’s paper “Algebraic $K$-Theory of Spaces”, at least to the point where we can give a detailed proof of the additivity theorem. Along the way we’ll see plenty of examples. Whereas Waldhausen was more interested in applying his framework directly to topology, we’ll concentrate on algebraic categories.

In this post, we’ll just introduce a weaker notion: categories with cofibrations. A Waldhausen category will then be a category of cofibrations together with a new class of maps, weak equivalences that we’ll only introduce later. Before we state the definition of a category with cofibrations, recall that a pointed category is a category with an object that is both initial and final. Whenever we have a pointed category, we’ll use the symbol $*$ to denote a choice of pointed object.

Definition. A category with cofibrations is a pointed category $\Ccl$ together with a subcategory ${\rm co}\Ccl$. The morphisms is ${\rm co}\Ccl$ will be denoted by $A\rightarrowtail B$ and they are required to satisfy:

  1. ${\rm co}\Ccl$ contains the isomorphisms.
  2. For each $A\in\Ccl$, the morphism $*\rightarrowtail A$ is a cofibration.
  3. (Cobase Change) For each cofibration $A\rightarrowtail B$ and each morphism $A\to C$ the pushout $B\cup_AC$ of the diagram $C\rightarrow A\rightarrowtail B$ exists and the morphism $C\rightarrowtail B\cup_AC$ is a cofibration.

Here are some basic examples and nonexamples of a category with cofibrations:

  1. An abelian category is a category with cofibrations, where the cofibrations are the injections.
  2. The category of rings is not a category of cofibrations, since it is not pointed.
  3. The category of finite sets is a category of cofibrations with the cofibrations being injections. This example is rather funny because proving the third “cobase change” axiom is ever so slightly more confusing than for abelian categories.
  4. The category of CW complexes having a given CW complex $X$ as a retract. Here, the cofibrations are the usual cofibrations in the category of CW complexes.

If Waldhausen categories are to be a good way of constructing algebraic $K$-theory, then they had better do at least the same thing as exact categories and the $Q$-construction. Recall that an exact category is an additive category $\Ccl$ with a distinguished class of sequences $\Ecl$ of the form $0\to A\to B\to C\to 0$ such that there is a fully faithful embedding of $\Ccl$ into an abelian category satisfying the two axioms:

  1. $\Ecl$ is exactly the set of all exact sequence in $\Acl$ that are contained in $\Ccl$
  2. If $0\to A\to B\to C\to 0$ is an exact sequence in $\Acl$ with $A$ and $C$ in $\Ccl$ then $B$ is isomorphic to some object in $\Ccl$.

The fundamental motivating example of an exact category is the category $\Pcl$ of finitely generated projective modules over a ring $R$, where the morphisms are the morphisms of $R$-modules and the exact sequences are just the exact sequences of $R$-modules, making the first axiom by definition. The second axiom is satisfied because an exact sequence $0\to A\to B\to C\to 0$ with $A$ and $C$ projective splits, giving an isomorphism $B\cong A\oplus C$, and $A\oplus C$ is projective. So in this case, $B$ is actually in $\Pcl$.

Given an exact category $\Ccl$, we can define a cofibration to be a monomorphism $A\to B$ that appears in some exact sequence $0\to A\to B\to C\to 0$ in the distinguished class of exact sequences $\Ecl$. Wait a minute, isn’t every monomorphism in $\Ccl$ admissible? No! For instance, consider $\Ccl$ to the smallest full additive subcategory of the category of abelian groups containing the objects $0$ and $\Z$. So this category has objects $\Z, \Z\oplus\Z, \Z\oplus\Z\oplus\Z,\dots$. This category is certainly not abelian, but is naturally embedded into an abelian one, viz. the category of abelian groups. The induced exact structure does not contain $\Z\to \Z$ given by multiplication by $n > 1$ as an admissible monic, since $\Ccl$ does not contain $\Z/n$. At any rate, if we do have an exact category, we can make it into a category with cofibrations as follows.

Theorem. Any exact category $\Ccl$ becomes a category with cofibrations once we choose the cofibrations to be the admissible monics.

The easiest axiom to verify is the first: if $A\to B$ is an isomorphism in $\Ccl$ then the sequence $0\to A\to B\to 0\to 0$ is certainly exact in any larger abelian category and contained in $\Ccl$ and so $A\to B$ is an admissible monic, hence a cofibration. What about $0\to A$ for any $A$? Again, the sequence $0\to 0\to A\to A\to 0$ where $A\to A = 1_A$ is certainly exact in any ambient abelian category, and also contained in $\Ccl$.

We still haven’t shown two things: that admissible monics are actually closed under composition, and the cobase change axiom. We shall do this in the next post, but you might like to try it first yourself.

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