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 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 be an additive category; that is, is enriched over abelian groups. For a pair of objects in , a biproduct of and (MacLane p.194) is an object together with maps which satisfy the following equalities:
If and are objects of , a biproduct of these objects (MacLane p.196) is an object together with maps for which satisfy the following equalities:
where if and if . Note that the equalities and are automatically verified in the case .
Let be a category with zero morphisms; that is, is enriched over pointed sets (which is notably the case when has a zero object). For a pair of objects in , suppose a product and a coproduct 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 the first morphisms are the projections and co-projections, respectively, and where is the zero morphism from to . Thus , where denotes the map induced by and .
If the morphism in def. , 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.
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 is an arbitrary category, without any assumption of pointedness, additivity, etc.
The biproduct of and is a tuple
such that is a product tuple, is a coproduct tuple, and
See Definition 3.1 in Karvonen 2020.
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 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.
The aforementioned result of Lack implies that a semiadditive category is a cocartesian object in the 2-category of cartesian monoidal categories and product-preserving functors. In fact, by Eckmann-Hilton, any monoidal structure defined on an object of such a 2-category has to coincide with the one given by the cartesian product of . Thus when we specify a cocartesian one on , Eckmann-Hilton gives us a natural isomorphism .
The exact same argument also shows semiadditive categories are cartesian objects in the 2-category of cocartesian monoidal categories and coproduct-preserving functors.
Given a category 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 with finite coproducts and products carries any natural isomorphism from coproducts to products, then
commutes for any two object and .
Hence is an isomorphism so that 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 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).
Conversely, if is already known to be enriched over commutative 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 commutative monoids must automatically preserve finite biproducts, so that finite biproducts are a type of Cauchy colimit (i.e. absolute colimit). Moreover, any product or coproduct in a category enriched over commutative 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.
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.
The category Rel of sets and relations between sets.
The category SupLat of sup-lattices.
Any compact closed category with products (or coproducts); see Houston 06, and for generalizations Garner-Schaeppi 15 and Zekic 21.
Some categories possess all binary products and coproducts but they are not biproducts and the category is thus not a semiadditive category. From above, we know that they are not enriched over the category of commutative monoids.
The category Set of sets and functions, where the product is given by the cartesian product of sets and the coproduct by the disjoint union of sets.
The category Top of topological spaces and continuous functions where the product is again given by the cartesian product and the coproduct by the disjoint union.
The category Grp of groups, where the product is given by the cartesian product and the coproduct by the free product of groups.
William Lawvere, around p. 56 of: Introduction to Linear Categories and Applications, course lecture notes (1992) [pdf, pdf]
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)
Richard Garner and Daniel Schaeppi, When coproducts are biproducts, Math. Proc. Camb. Phil. Soc. 161 (2016) 47-51, arXiv:1505.01669, 2015
Mladen ZekiΔ, Biproducts in monoidal categories, Publications de LβInstitut Mathematique, DOI, 2021
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 (2020) 229β238 arXiv:1801.06488, journal pdf
A related discussion is archived at Forum.
Last revised on March 6, 2024 at 00:31:36. See the history of this page for a list of all contributions to it.