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.

Idea

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.

Definitions

In an additive category

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

$$i_{1};p_{1}=1_{a}, \quad i_{2};p_{2}=1_{b}, \quad p_{1};i_{1} + p_{2};i_{2} = 1_{a \oplus b}$$

If $n \ge 3$ and $a_1,...,a_n$ are objects of $\mathcal{C}$, a biproduct of these objects (MacLane p.196) is an object $\underset{1 \le j \le n}{\bigoplus}a_j$ together with maps for $1 \le k \le n$ which satisfy the following equalities:

$$i_{k};p_{l}=\delta_{k,l}, \quad \underset{1 \le k \le n}{\sum}p_k;i_k=1_{\underset{1 \le j \le n}{\bigoplus}a_j}$$

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

In a category with zero morphisms

Let $\mathcal{C}$ be a category with zero morphisms; that is, $\mathcal{C}$ is enriched over pointed sets (which is notably the case when $\mathcal{C}$ has a zero object). For $c_1, c_2$ a pair of objects in $\mathcal{C}$, suppose a product $c_1 \times c_2$ and a coproduct $c_1 \sqcup c_2$ both exist.

Definition

Write

$$r_{c_1,c_2} \;\colon\; c_1 \sqcup c_2 \longrightarrow c_1 \times c_2$$

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

(1)$$\left( c_i \longrightarrow c_1 \sqcup c_2 \overset{\; r \;}{\longrightarrow} c_1 \times c_2 \longrightarrow c_j \right) = \left\{ \array{ Id_{c_i} & if & i = j \\ 0_{i,j} & if & i \neq j } \right. \,$$

where the last and the first morphisms are the projections and co-projections, respectively, and where $0_{i,j}$ 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 $(f, g) \colon d \to a \times b$ denotes the map induced by $f \colon d \to a$ and $g \colon d \to b$.

Definition

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

$$r \;\colon\; (-)\sqcup (-) \stackrel{\simeq}{\longrightarrow} (-) \times (-)$$

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

Remark

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 $C$ is an arbitrary category, without any assumption of pointedness, additivity, etc.

The biproduct of $c_1$ and $c_2$ is a tuple

$$(c_1\oplus c_2,p_1:c_1\oplus c_2\to c_1,p_2:c_1\oplus c_2\to c_2,i_1:c_1\to c_1\oplus c_2,i_2:c_2\to c_1\oplus c_2)$$

such that $(c_1\oplus c_2,p_1,p_2)$ is a product tuple, $(c_1\oplus c_2,i_1,i_2)$ is a coproduct tuple, and

$$p_1 i_1=id,$$

$$p_2 i_2=id,$$

$$i_1 p_1 i_2 p_2 = i_2 p_2 i_1 p_1.$$

See Definition 3.1 in Karvonen 2020.

Semiadditive categories

A category $C$ 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 $c_1 \sqcup c_2 \cong c_1 \times c_2$ — 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 $\mathcal{C}$ of such a 2-category has to coincide with the one given by the cartesian product of $\mathcal{C}$. Thus when we specify a cocartesian one on $\mathcal{C}$, Eckmann-Hilton gives us a natural isomorphism $c_1 \sqcup c_2 \cong c_1 \times c_2$.

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

Properties

Semiadditivity as structure/property

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

$$\psi_{(-),(-)} : (-)\sqcup (-) \stackrel{\simeq}{\longrightarrow} (-)\times(-) \,.$$

Proposition

(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

$$\array { c_1\sqcup c_2 & \overset{\psi_{c_1, 0} + \psi_{0, c_2}}\rightarrow & c_1 \sqcup c_2 \\ & \searrow^{r_{c_1, c_2}} & \downarrow^{\psi_{c_1, c_2}} \\ & & c_1 \times c_2 }$$

commutes for any two object $c_1$ 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.

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 $f, g: a \to b$ in $C$, let their sum $f + g: a \to b$ be

$$a \to a \times a \cong a \oplus a \overset{f \oplus g}{\to} b \oplus b \cong b \sqcup b \to b .$$

One proves that $+$ is associative and commutative. Of course, the zero morphism $0: a \to b$ is the usual zero morphism given by the zero object:

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

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).

Biproducts as enriched Cauchy colimits

Conversely, if $C$ is already known to be enriched over commutative 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 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 $n_1 p_1 + n_2 p_2 = 1_{c_1\oplus c_2}$ replaced by $\bigvee_{i} n_i p_i = 1_{\bigoplus_i c_i}$. In particular, the category of suplattices has all small biproducts.

Biproducts from duals

The existence of dual objects tends to imply (semi)additivity; see (Houston 08, MO discussion).

Examples

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.

• K(n)-local stable homotopy theory (Hopkins-Lurie 14)

• 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.

Non-examples

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.

Related concepts

• bilimit

• semiadditive (∞,1)-category

• pointed category

References

• Saunders MacLane, Categories for the working mathematician

• 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 Zekić, Biproducts in monoidal categories, Publications de L’Institut 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 géométrie différentielle catégoriques 61 3 (2020) 229–238 $[$arXiv:1801.06488, journal pdf$]$

A related discussion is archived at $n$Forum.