category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A category $C$ with finite products $(-)\times(-)$ and finite coproducts $(-) + (-)$ is called (finitary) distributive if for any $X,Y,Z\in C$ the canonical distributivity morphism
is an isomorphism. The canonical morphism is the unique morphism such that $X\times Y \to X\times (Y+Z)$ is $X\times i$, where $i\colon Y\to Y +Z$ is the coproduct injection, and dually for $X\times Z \to X\times (Y+Z)$.
This notion is part of a hierarchy of distributivity for monoidal structures, and generalizes to distributive monoidal categories and rig categories. A linearly distributive category is not distributive in this sense.
This axiom on binary coproducts easily implies the analogous $n$-ary result for $n\gt 2$. In fact it also implies the analogous 0-ary statement that the projection
is an isomorphism for any $X$ (see Proposition below). Moreover, for a category with finite products and coproducts to be distributive, it actually suffices for there to be any natural family of isomorphisms $X\times Y + X\times Z \cong X\times (Y+Z)$, not necessarily the canonical ones (Lack).
A category $C$ with finite products and all small coproducts is infinitary distributive if the statement applies to all small coproducts. One can also consider $\kappa$-distributivity for a cardinal number $\kappa$, meaning the statement applies to coproducts of cardinality $\lt\kappa$.
Any extensive category with finite products is distributive, but the converse is not true.
In a category with products and coproducts, if products distribute over binary coproducts, then coproduct coprojections are monic.
Let $i_B: B \to B + C$ be a coproduct coprojection, and suppose given maps $f, g: A \to B$ such that $i_B f = i_B g$. We observe that the coprojection
is monic because it has a retraction $(1_{A \times B}, \phi): A \times B + A \times C \to A \times B$. (All we need here is the existence of a map $\phi: A \times C \to A \times B$, for example the composite $A \times C \stackrel{\pi_A}{\to} A \stackrel{\langle 1_A, f \rangle}{\to} A \times B$.)
The composite of the coprojection $i$ with the canonical isomorphism $A \times B + A \times C \cong A \times (B + C)$, namely $1_A \times i_B: A \times B \to A \times (B + C)$, is therefore also monic. Given that $\langle 1_A, i_B f \rangle = \langle 1_A, i_B g \rangle: A \to A \times (B + C)$, we conclude
whence $\langle 1_A, f\rangle = \langle 1_A, g\rangle: A \to A \times B$ since $1_A \times i_B$ is monic. It follows that $f = g$, as was to be shown.
If products distribute over binary coproducts, then products distribute over nullary coproducts (i.e., the projection $X \times 0 \to 0$ is an isomorphism for all objects $X$).
We show that $X \times 0$ is initial. Clearly $\hom(X \times 0, Y)$ is inhabited by $X \times 0 \to 0 \to Y$ for any object $Y$. On the other hand, since the two coprojections $0 \to 0 + 0$ coincide, the same holds for the two coprojections $X \times 0 \to (X \times 0) + (X \times 0)$, by applying the distributivity isomorphism $X \times (0 + 0) \cong (X \times 0) + (X \times 0)$. This is enough to show that any two maps $X \times 0 \to Y$ coincide, since given maps $f, g : X \times 0 \to Y$, we have $f = [f, g] \circ i_1 = [f, g] \circ i_2 = g$.
In a distributive category, the initial object is strict.
Given an arrow $f: A \to 0$, we have that $\pi_A: A \times 0 \to A$ is a retraction of $\langle 1, f \rangle: A \to A \times 0$, so that $A$ is a retract of $A \times 0 \cong 0$. But retracts of initial objects are initial.
For example:
any topos,
the category Top of topological spaces with respect to forming product topological spaces and disjoint union topological spaces;
are distributive categories (hence distributive monoidal categories, hence rig categories).
These categories have in common that they are extensive. An example^{1} of a distributive category that is not extensive is given by
Since for posets viewed as categories, finite products and coproducts are given by meets and joins we find:
A further non-example:
Aurelio Carboni, Stephen Lack, R. F. C. Walters, Introduction to extensive and distributive categories, JPAA 84 (1993) pp. 145-158 (doi:10.1016/0022-4049(93)90035-R)
Stephen Lack, Non-canonical isomorphisms. arXiv:0912.2126.
Pointed out by Peter Freyd in this discussion. ↩
Last revised on June 8, 2022 at 15:04:08. See the history of this page for a list of all contributions to it.