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
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
A biproduct in a category $\mathcal{C}$ 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 necessarily additive, is called a semiadditive category.
Let $\mathcal{C}$ 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 $c_1 \times c_2$ and a coproduct $c_1 \sqcup c_2$ both exist.
Write
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 $0_{i,j}$ is the zero morphism from $c_i$ to $c_j$. Thus $r_{c_1, c_2} = (Id_{c_1}, 0_{1,2}) \sqcup (0_{2,1}, Id_{c_2})$, where $(f, g): d \to a \times b$ denotes the map induced by $f : d \to a$ and $g : d \to b$.
If the morphism $r_{c_1,c_2}$ in def. , is an isomorphism, then the isomorphic objects $c_1 \times c_2$ and $c_1 \sqcup c_2$ 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).
If $r_{c_1,c_2}$ is an isomorphism for all objects $c_1, c_2 \in \mathcal{C}$ and hence a natural isomorphism
then $\mathcal{C}$ is called a semiadditive category.
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.
Suppose $C$ is an arbitrary category, without any assumption of pointedness, additivity, etc.
The biproduct of $c_1$ and $c_2$ is a tuple
such that $(c_1\oplus c_2,p_1,p_2)$ is a product tuple, $(c_1\oplus c_2,i_1,i_2)$ is a coproduct tuple, and
See Definition 3.1 in Karvonen.
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 $c_1 \sqcup c_2 \to c_1 \times c_2$ 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 $c_1 \sqcup c_2 \cong c_1 \times c_2$ β 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 β Biproducts imply enrichment.
Given a category $\mathcal{C}$ with zero morphisms, one may imagine equipping it with the structure of a chosen natural isomorphism
(Lack 09, proof of theorem 5). If a category $\mathcal{C}$ with finite coproducts and products carries any natural isomorphism $\psi_{(-),(-)}$ from coproducts to products, then
commutes for any two object $c_1$ and $c_2$.
Hence $r_{c_1, c_2}$ is an isomorphism so that $\mathcal{C}$ is semi-additive. See non-canonical isomorphism for more.
A semiadditive category is automatically enriched over the monoidal category of commutative monoids with the usual tensor product, as follows.
Given two morphisms $f, g: a \to b$ in $C$, let their sum $f + g: a \to b$ be
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:
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 $Ab$ 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 $SLat$ of semilattices (and is therefore a kind of 2-poset).
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 $p_j n_i = \delta_{i j}$ (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 $\bigvee_{i} n_i p_i = 1_{\bigoplus_i c_i}$. In particular, the category of suplattices has all small biproducts.
The existence of dual objects tends to imply (semi)additivity; see (Houston 08, MO discussion).
Categories with biproducts include:
The category Ab of abelian groups. More generally, any abelian category.
The category of (finitely generated) projective modules over a given ring.
Any triangulated category, in particular the derived category of a ring, or the homotopy category of spectra.
The category of abelian categories and exact functors.
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)
Martti Karvonen, Biproducts without pointedness, Cahiers de topologie et gΓ©omΓ©trie diffΓ©rentielle catΓ©goriques, 61(3):229β238, 2020. arXiv.
A related discussion is archived at $n$Forum.