category theory

Idea

A biproduct in a category $C$ is an operation that is both a product and a coproduct, in a compatible way. Finite biproducts are best known from additive categories and their generalisations.

Morphisms between finite biproducts are encoded in a matrix calculus.

Definition

Let $C$ be a category with zero morphisms; that is, $C$ is enriched over pointed sets (for example, $C$ might have a zero object). For ${c}_{1},{c}_{2}$ two objects in $C$, suppose a product ${c}_{1}×{c}_{2}$ and a coproduct ${c}_{1}\bigsqcup {c}_{2}$ both exist. Then consider the canonical morphism

$r:{c}_{1}\bigsqcup {c}_{2}\to {c}_{1}×{c}_{2}$r : c_1 \sqcup c_2 \to c_1 \times c_2

defined by

$\left({c}_{i}\to {c}_{1}\bigsqcup {c}_{2}\stackrel{r}{\to }{c}_{1}×{c}_{2}\to {c}_{j}\right)=\left\{\begin{array}{cc}{\mathrm{Id}}_{{c}_{i}}& \mathrm{if}i=j\\ {0}_{i,j}& \mathrm{if}i\ne j\end{array}\phantom{\rule{thinmathspace}{0ex}}$\left( c_i \to c_1 \sqcup c_2 \stackrel{r}{\to} c_1 \times c_2 \to c_j \right) = \left\{ \array{ Id_{c_i} & if i = j \\ 0_{i,j} & if i \neq j } \right. \,

where ${0}_{i,j}$ is the zero morphism from ${c}_{i}$ to ${c}_{j}$.

If this morphism $r$ is an isomorphism, then the isomorphic objects ${c}_{1}×{c}_{2}$ and ${c}_{1}\bigsqcup {c}_{2}$ are called the biproduct of ${c}_{1}$ and ${c}_{2}$. This object is often denoted ${c}_{1}\oplus {c}_{2}$, alluding to the direct sum (which is often an example).

The above definition has a straightforward generalization to biproducts of any number of objects (although this requires extra structure on the category in constructive mathematics if the set indexing these objects might not have decidable equality). A zero object is the biproduct of no objects.

Mike: Can anyone give a definition of a biproduct that doesn’t require the category to be presupposed to have a zero object, but which specializes to a zero object in the 0-ary case?

Toby Bartels: Actually, our current definition does this; I just wrote it badly. Of course, now the category starts with an enriched structure … but at least it is a weaker requirement.

Mike: The reason I ask is that I’m trying to work out an indexed version of biproducts. In a fibration or indexed category, coproducts are left adjoint ${f}_{!}$ to reindexing ${f}^{*}$ and products are right adjoint ${f}_{*}$, so having $f$-indexed biproducts should mean that some canonical map ${f}_{!}\to {f}_{*}$ is an isomorphism. But I’m not having much luck constructing such a canonical map. Perhaps this is related to the need for decidable equality?

Toby: If it helps, here is a bit more on that subject, which I wrote at direct sum:

An arbitrary index set will still work if $C$ is enriched over the category of sets and partial functions; this may be embedded as a full subcategory of the category of pointed sets, and the embedding is an equivalence of categories if and only if the law of excluded middle holds. But the usual examples of $C$ are not (constructively) so enriched.

Mike: Was that what you had in mind for the “extra structure” above? It occurred to me that it could also mean the existence of “subzero objects,” i.e. ${\coprod }_{A}X\to {\prod }_{A}X$ is an isomorphism whenever $A$ is a subsingleton—which is also, I think, not constructively true in the usual examples.

Toby: Yes, that's what I had in mind. (Although you could also do a more local version, letting the requirement that the index set be discrete and the requirement the hom-sets be enriched over ${\mathrm{Set}}_{\mathrm{part}}$ meet halfway, if you see what I mean.)

A category $C$ with all finite biproducts is called a semiadditive category. More precisely, this means that $C$ has all finite products and coproducts, that the unique map $0\to 1$ is an isomorphism (hence $C$ has a zero object), and that the canonical maps ${c}_{1}\bigsqcup {c}_{2}\to {c}_{1}×{c}_{2}$ defined above are isomorphisms.

Amusingly, for $C$ to be semiadditive, it actually suffices to assume that $C$ has finite products and coproducts and that there exists any natural family of isomorphisms ${c}_{1}\bigsqcup {c}_{2}\cong {c}_{1}×{c}_{2}$ — not necessarily the canonical maps constructed above. A proof can be found in

An additive category, although normally defined through the theory of enriched categories, may also be understood as a semiadditive category with an extra property, as explained below.

The existence of duals also tends to imply (semi)additivity; see this paper and this MO question.

Biproducts imply enrichment

A semiadditive category is automatically enriched over the monoidal category of abelian monoids with the usual tensor product, as follows.

Given two morphisms $f,g:a\to b$ in $C$, let their sum $f+g:a\to b$ be

$a\to a×a\cong a\oplus a\stackrel{f\oplus g}{\to }b\oplus b\cong b\bigsqcup b\to b.$a \to a \times a \cong a \oplus a \overset{f \oplus g}{\to} b \oplus b \cong b \sqcup b \to b .

One proves that $+$ is associative and commutative. Of course, the zero morphism $0:a\to b$ is the usual zero morphism given by the zero object:

$a\to 1\cong 0\to b.$a \to 1 \cong 0 \to b .

One proves that $0$ is the neutral element for $+$ and that this matches the $0$ morphism that we began with in the definition. Note that in addition to a zero object, this construction actually only requires biproducts of an object with itself, i.e. biproducts of the form $a\oplus a$ rather than the more general $a\oplus b$.

If additionally every morphism $f:a\to b$ has an inverse $-f:a\to b$, then $C$ is enriched over the category $\mathrm{Ab}$ of abelian groups and is therefore (precisely) an additive category.

If, on the other hand, the addition of morphisms is idempotent ($f+f=f$), then $C$ is enriched over the category $\mathrm{SLat}$ of semilattices (and is therefore a kind of 2-poset).

Biproducts as enriched Cauchy colimits

Conversely, if $C$ is already known to be enriched over abelian monoids, then a binary biproduct may be defined purely diagrammatically as an object ${c}_{1}\oplus {c}_{2}$ together with injections ${n}_{i}:{c}_{i}\to {c}_{1}\oplus {c}_{2}$ and projections ${p}_{i}:{c}_{1}\oplus {c}_{2}\to {c}_{i}$ such that ${p}_{j}{n}_{i}={\delta }_{ij}$ (the Kronecker delta?) and ${n}_{1}{p}_{1}+{n}_{2}{p}_{2}={1}_{{c}_{1}\oplus {c}_{2}}$. It is easy to check that makes ${c}_{1}\oplus {c}_{2}$ a biproduct, and that any binary biproduct must be of this form. Similarly, an object $z$ of such a category is a zero object precisely when ${1}_{z}={0}_{z}$, its identity is equal to the zero morphism. It follows that functors enriched over abelian monoids must automatically preserve finite biproducts, so that finite biproducts are a type of Cauchy colimit. Moreover, any product or coproduct in a category enriched over abelian monoids is actually a biproduct.

For categories enriched over suplattices, this extends to all small biproducts, with the condition ${n}_{1}{p}_{1}+{n}_{2}{p}_{2}={1}_{{c}_{1}\oplus {c}_{2}}$ replaced by ${\bigvee }_{i}{n}_{i}{p}_{i}={1}_{{⨁}_{i}{c}_{i}}$. In particular, the category of suplattices has all small biproducts.

Revised on September 2, 2010 08:31:58 by Urs Schreiber (87.212.203.135)