nLab
biproduct

This entry is about coproducts coinciding with products. For the notion of biproduct in the sense of bicategory theory see at 2-limit. See at bilimit for general disambiguation.

Context

Additive and abelian categories

Category theory

Limits and colimits

Contents

Idea

A biproduct in a category 𝒞\mathcal{C} is an operation that is both a product and a coproduct, in a compatible way. Morphisms between finite biproducts are encoded in a matrix calculus.

Finite biproducts are best known from additive categories. A category which has biproducts but is not necessarily enriched in Ab, hence not necessatily additive, is called a semiadditive category.

Definition

Let 𝒞\mathcal{C} be a category with zero morphisms; that is, CC is enriched over pointed sets (for example, CC might have a zero object). For c 1,c 2c_1, c_2 two objects in CC, suppose a product c 1×c 2c_1 \times c_2 and a coproduct c 1c 2c_1 \sqcup c_2 both exist.

Definition

Write

r c 1,c 2:c 1c 2c 1×c 2 r_{c_1,c_2} : c_1 \sqcup c_2 \to c_1 \times c_2

for the morphism which is uniquely defined (via the universal property of coproduct and product) by the condition that

(c ic 1c 2rc 1×c 2c j)={Id c i ifi=j 0 i,j ifij \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 the last and first morphisms are the projections and co-projections, respectively, and where 0 i,j0_{i,j} is the zero morphism from c ic_i to c jc_j.

Definition

If the morphism r c 1,c 2r_{c_1,c_2} in def. 1, is an isomorphism, then the isomorphic objects c 1×c 2c_1 \times c_2 and c 1c 2c_1 \sqcup c_2 are called the biproduct of c 1c_1 and c 2c_2. This object is often denoted c 1c 2c_1 \oplus c_2, alluding to the direct sum (which is often an example).

If r c 1,c 2r_{c_1,c_2} is an isomorphism for all objects c 1,c 2𝒞c_1, c_2 \in \mathcal{C} and hence a natural isomorphism

r:()()()×() r \;\colon\; (-)\coprod (-) \stackrel{\simeq}{\longrightarrow} (-) \times (-)

then 𝒞\mathcal{C} is called a semiadditive category.

Remark

Definition 2 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.

Semiadditive categories

A category CC with all finite biproducts is called a semiadditive category. More precisely, this means that CC has all finite products and coproducts, that the unique map 010\to 1 is an isomorphism (hence CC has a zero object), and that the canonical maps c 1c 2c 1×c 2c_1 \sqcup c_2 \to c_1 \times c_2 defined above are isomorphisms.

Amusingly, for CC to be semiadditive, it actually suffices to assume that CC has finite products and coproducts and that there exists any natural family of isomorphisms c 1c 2c 1×c 2c_1 \sqcup c_2 \cong c_1 \times c_2 — not necessarily the canonical maps constructed above. A proof can be found in (Lack 09).

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 at Properties – Biproducts imply enrichment.

Properties

Semiadditivity as structure/property

Given a category 𝒞\mathcal{C} with zero morphism, one may imagine equipping it with the structure of a chosen natural isomorphism

()()()×(). (-)\coprod (-) \stackrel{\simeq}{\longrightarrow} (-)\times(-) \,.
Proposition

If a category with finite coproducts and products carries any natural isomorphism between coproducts and products, then it is semi-additive.

(Lack 09, theorem 5).

Biproducts imply enrichment – Relation to additive categories

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

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

aa×aaafgbbbbb. 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:ab0: a \to b is the usual zero morphism given by the zero object:

a10b. a \to 1 \cong 0 \to b .

One proves that 00 is the neutral element for ++ and that this matches the 00 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 aaa\oplus a rather than the more general aba\oplus b.

If additionally every morphism f:abf: a \to b has an inverse f:ab-f: a \to b, then CC is enriched over the category AbAb of abelian groups and is therefore (precisely) an additive category.

If, on the other hand, the addition of morphisms is idempotent (f+f=ff+f=f), then CC is enriched over the category SLatSLat of semilattices (and is therefore a kind of 2-poset).

Biproducts as enriched Cauchy colimits

Conversely, if CC is already known to be enriched over abelian monoids, then a binary biproduct may be defined purely diagrammatically as an object c 1c 2c_1\oplus c_2 together with injections n i:c ic 1c 2n_i:c_i\to c_1\oplus c_2 and projections p i:c 1c 2c ip_i:c_1\oplus c_2 \to c_i such that p jn i=δ ijp_j n_i = \delta_{i j} (the Kronecker delta) and n 1p 1+n 2p 2=1 c 1c 2n_1 p_1 + n_2 p_2 = 1_{c_1\oplus c_2}. It is easy to check that makes c 1c 2c_1\oplus c_2 a biproduct, and that any binary biproduct must be of this form. Similarly, an object zz of such a category is a zero object precisely when 1 z=0 z1_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 1p 1+n 2p 2=1 c 1c 2n_1 p_1 + n_2 p_2 = 1_{c_1\oplus c_2} replaced by in ip i=1 ic i\bigvee_{i} n_i p_i = 1_{\bigoplus_i c_i}. In particular, the category of suplattices has all small biproducts.

Biproducts from duals

The existence of dual objects tends to imply (semi)additivity; see ({Houston 06}, MO discussion).

Examples

Categories with biproducts include:

References

A related discussion is archived at nnForum.

Revised on February 1, 2014 09:03:57 by Urs Schreiber (89.204.130.13)