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.
Additive and abelian categories
Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
A biproduct in a category 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.
Let be a category with zero morphisms; that is, is enriched over pointed sets (for example, might have a zero object). For two objects in , suppose a product and a coproduct both exist.
for the morphism which is uniquely defined (via the universal property of coproduct and product) by the condition that
where the last and first morphisms are the projections and co-projections, respectively, and where is the zero morphism from to .
If the morphism in def. 1, is an isomorphism, then the isomorphic objects and are called the biproduct of and . This object is often denoted , alluding to the direct sum (which is often an example).
If is an isomorphism for all objects and hence a natural isomorphism
then is called a semiadditive category.
A category with all finite biproducts is called a semiadditive category. More precisely, this means that has all finite products and coproducts, that the unique map is an isomorphism (hence has a zero object), and that the canonical maps defined above are isomorphisms.
Amusingly, for to be semiadditive, it actually suffices to assume that has finite products and coproducts and that there exists any natural family of isomorphisms — 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.
Semiadditivity as structure/property
Given a category with zero morphism, one may imagine equipping it with the structure of a chosen natural isomorphism
(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 in , let their sum be
One proves that is associative and commutative. Of course, the zero morphism is the usual zero morphism given by the zero object:
One proves that is the neutral element for and that this matches the 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 rather than the more general .
If additionally every morphism has an inverse , then is enriched over the category of abelian groups and is therefore (precisely) an additive category.
If, on the other hand, the addition of morphisms is idempotent (), then is enriched over the category of semilattices (and is therefore a kind of 2-poset).
Biproducts as enriched Cauchy colimits
Conversely, if is already known to be enriched over abelian monoids, then a binary biproduct may be defined purely diagrammatically as an object together with injections and projections such that (the Kronecker delta) and . It is easy to check that makes a biproduct, and that any binary biproduct must be of this form. Similarly, an object of such a category is a zero object precisely when , 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 replaced by . 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 08, MO discussion).
Categories with biproducts include:
Stephen Lack, Non-canonical isomorphisms, (arXiv:0912.2126).
Robin Houston, Finite Products are Biproducts in a Compact Closed Category, Journal of Pure and Applied Algebra, Volume 212, Issue 2, February 2008, Pages 394-400 (arXiv:math/0604542)
Michael Hopkins, Jacob Lurie, Ambidexterity in K(n)-Local Stable Homotopy Theory (2014)
A related discussion is archived at Forum.