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.


Additive and abelian categories

Category theory

Limits and colimits



A biproduct in a category π’ž\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.


In an additive category

Let π’ž\mathcal{C} be an additive category; that is, π’ž\mathcal{C} is enriched over abelian groups. For a,ba, b a pair of objects in π’ž\mathcal{C}, a biproduct of aa and bb (MacLane p.194) is an object aβŠ•ba \oplus b together with maps which satisfy the following equalities:

i 1;p 1=1 a,i 2;p 2=1 b,p 1;i 1+p 2;i 2=1 aβŠ•b i_{1};p_{1}=1_{a}, \quad i_{2};p_{2}=1_{b}, \quad p_{1};i_{1} + p_{2};i_{2} = 1_{a \oplus b}

If nβ‰₯3n \ge 3 and a 1,...,a na_1,...,a_n are objects of π’ž\mathcal{C}, a biproduct of these objects (MacLane p.196) is an object ⨁1≀j≀na j\underset{1 \le j \le n}{\bigoplus}a_j together with maps for 1≀k≀n1 \le k \le n which satisfy the following equalities:

i k;p l=Ξ΄ k,l,βˆ‘1≀k≀np k;i k=1 ⨁1≀j≀na j i_{k};p_{l}=\delta_{k,l}, \quad \underset{1 \le k \le n}{\sum}p_k;i_k=1_{\underset{1 \le j \le n}{\bigoplus}a_j}

where δ k,l=0 a k,a l\delta_{k,l}=0_{a_k,a_l} if k≠lk \neq l and δ k,l=1 a k\delta_{k,l}=1_{a_k} if k=lk = l. Note that the equalities i 1;p 2=0i_1;p_2=0 and i 2;p 1=0i_2;p_1=0 are automatically verified in the case n=2n=2.

In a category with zero morphisms

Let π’ž\mathcal{C} be a category with zero morphisms; that is, π’ž\mathcal{C} is enriched over pointed sets (which is notably the case when π’ž\mathcal{C} has a zero object). For c 1,c 2c_1, c_2 a pair of objects in π’ž\mathcal{C}, suppose a product c 1Γ—c 2c_1 \times c_2 and a coproduct c 1βŠ”c 2c_1 \sqcup c_2 both exist.



r c 1,c 2:c 1βŠ”c 2⟢c 1Γ—c 2 r_{c_1,c_2} \;\colon\; c_1 \sqcup c_2 \longrightarrow c_1 \times c_2

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

(1)(c i⟢c 1βŠ”c 2⟢rc 1Γ—c 2⟢c j)={Id c i if i=j 0 i,j if iβ‰ j \left( c_i \longrightarrow c_1 \sqcup c_2 \overset{\; r \;}{\longrightarrow} c_1 \times c_2 \longrightarrow c_j \right) = \left\{ \array{ Id_{c_i} & if & i = j \\ 0_{i,j} & if & i \neq j } \right. \,

where the last and the 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. Thus r c 1,c 2=(Id c 1,0 1,2)βŠ”(0 2,1,Id c 2)r_{c_1, c_2} = (Id_{c_1}, 0_{1,2}) \sqcup (0_{2,1}, Id_{c_2}), where (f,g):dβ†’aΓ—b(f, g) \colon d \to a \times b denotes the map induced by f:dβ†’af \colon d \to a and g:dβ†’bg \colon d \to b.


If the morphism r c 1,c 2r_{c_1,c_2} in def. , is an isomorphism, then the isomorphic objects c 1Γ—c 2c_1 \times c_2 and c 1βŠ”c 2c_1 \sqcup c_2 are called the biproduct of c 1c_1 and c 2c_2. This object is often denoted c 1βŠ•c 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 2βˆˆπ’žc_1, c_2 \in \mathcal{C} and hence a natural isomorphism

r:(βˆ’)βŠ”(βˆ’)βŸΆβ‰ƒ(βˆ’)Γ—(βˆ’) r \;\colon\; (-)\sqcup (-) \stackrel{\simeq}{\longrightarrow} (-) \times (-)

then π’ž\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.

Point-free definition

Suppose CC is an arbitrary category, without any assumption of pointedness, additivity, etc.

The biproduct of c 1c_1 and c 2c_2 is a tuple

(c 1βŠ•c 2,p 1:c 1βŠ•c 2β†’c 1,p 2:c 1βŠ•c 2β†’c 2,i 1:c 1β†’c 1βŠ•c 2,i 2:c 2β†’c 1βŠ•c 2)(c_1\oplus c_2,p_1:c_1\oplus c_2\to c_1,p_2:c_1\oplus c_2\to c_2,i_1:c_1\to c_1\oplus c_2,i_2:c_2\to c_1\oplus c_2)

such that (c 1βŠ•c 2,p 1,p 2)(c_1\oplus c_2,p_1,p_2) is a product tuple, (c 1βŠ•c 2,i 1,i 2)(c_1\oplus c_2,i_1,i_2) is a coproduct tuple, and

p 1i 1=id,p_1 i_1=id,
p 2i 2=id,p_2 i_2=id,
i 1p 1i 2p 2=i 2p 2i 1p 1.i_1 p_1 i_2 p_2 = i_2 p_2 i_1 p_1.

See Definition 3.1 in Karvonen 2020.

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 0β†’10\to 1 is an isomorphism (hence CC has a zero object), and that the canonical maps c 1βŠ”c 2β†’c 1Γ—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 1βŠ”c 2β‰…c 1Γ—c 2c_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 – Biproducts imply enrichment.

As (co)cartesian objects

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 π’ž\mathcal{C} of such a 2-category has to coincide with the one given by the cartesian product of π’ž\mathcal{C}. Thus when we specify a cocartesian one on π’ž\mathcal{C}, Eckmann-Hilton gives us a natural isomorphism c 1βŠ”c 2β‰…c 1Γ—c 2c_1 \sqcup c_2 \cong c_1 \times c_2.

The exact same argument also shows semiadditive categories are cartesian objects in the 2-category of cocartesian monoidal categories and coproduct-preserving functors.


Semiadditivity as structure/property

Given a category π’ž\mathcal{C} with zero morphisms, one may imagine equipping it with the structure of a chosen natural isomorphism

ψ (βˆ’),(βˆ’):(βˆ’)βŠ”(βˆ’)βŸΆβ‰ƒ(βˆ’)Γ—(βˆ’). \psi_{(-),(-)} : (-)\sqcup (-) \stackrel{\simeq}{\longrightarrow} (-)\times(-) \,.

(Lack 09, proof of theorem 5). If a category π’ž\mathcal{C} with finite coproducts and products carries any natural isomorphism ψ (βˆ’),(βˆ’)\psi_{(-),(-)} from coproducts to products, then

c 1βŠ”c 2 β†’Οˆ c 1,0+ψ 0,c 2 c 1βŠ”c 2 β†˜ r c 1,c 2 ↓ ψ c 1,c 2 c 1Γ—c 2 \array { c_1\sqcup c_2 & \overset{\psi_{c_1, 0} + \psi_{0, c_2}}\rightarrow & c_1 \sqcup c_2 \\ & \searrow^{r_{c_1, c_2}} & \downarrow^{\psi_{c_1, c_2}} \\ & & c_1 \times c_2 }

commutes for any two object c 1c_1 and c 2c_2.

Hence r c 1,c 2r_{c_1, c_2} is an isomorphism so that π’ž\mathcal{C} is semi-additive. See non-canonical isomorphism for more.

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:a→bf, g: a \to b in CC, let their sum f+g:a→bf + g: a \to b be

aβ†’aΓ—aβ‰…aβŠ•aβ†’fβŠ•gbβŠ•bβ‰…bβŠ”bβ†’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→b0: a \to b is the usual zero morphism given by the zero object:

a→1≅0→b. 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 aβŠ•aa\oplus a rather than the more general aβŠ•ba\oplus b.

If additionally every morphism f:aβ†’bf: a \to b has an inverse βˆ’f:aβ†’b-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 commutative monoids, then a binary biproduct may be defined purely diagrammatically as an object c 1βŠ•c 2c_1\oplus c_2 together with injections n i:c iβ†’c 1βŠ•c 2n_i:c_i\to c_1\oplus c_2 and projections p i:c 1βŠ•c 2β†’c ip_i:c_1\oplus c_2 \to c_i such that p jn i=Ξ΄ ijp_j n_i = \delta_{i j} (the Kronecker delta) and n 1p 1+n 2p 2=1 c 1βŠ•c 2n_1 p_1 + n_2 p_2 = 1_{c_1\oplus c_2}. It is easy to check that makes c 1βŠ•c 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 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 n 1p 1+n 2p 2=1 c 1βŠ•c 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 08, MO discussion).


Categories with biproducts include:


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.


A related discussion is archived at nnForum.

Last revised on March 6, 2024 at 00:31:36. See the history of this page for a list of all contributions to it.