nLab strict initial object

Idea

The empty set, among all sets, has two characteristic properties:

1. there is a unique function out of the empty set, to any other set;

2. there is no function to the empty set, except from itself.

The first property generalizes to arbitrary categories as the property of an initial object.

The corresponding generalization including also the second property is that of a strict initial object:

Definition

An initial object $\varnothing$ is called a strict initial object if every morphism to $\varnothing$ is an isomorphism:

(1)$$X \overset{f}{\longrightarrow} \varnothing \;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\; X \underoverset{\simeq}{f}{\longrightarrow} \emptyset \,.$$

Properties

• The Cartesian product of any object $X$ with a strict initial object is isomorphic to the strict initial object, $X \times \varnothing \simeq \varnothing$, because the projection $pr_1 \colon \varnothing \times X \to \varnothing$ exists by definition of Cartesian products, whence (1) implies that it is an isomorphism

  $$\varnothing \times X \underoverset {\;\;\;\simeq\;\;\;} {pr_1} {\longrightarrow} \varnothing \,.$$

• Strict initial objects may be understood as van Kampen colimits, see e.g. Sobocinski & Heindel (2011), Exp. 4.5 (i). Indeed, the van Kampen-property in this case requires that the slice category $C/\varnothing$ be equivalent to the terminal category $\mathbf{1}$.

Examples

• The empty set is a strict initial object in Sets.

• The empty topological space is a strict initial object in TopologicalSpaces.

• The empty groupoid is a strict initial object in Groupoids.

• The empty simplicial set is a strict initial object in SimplicialSets.

• The initial objects of any of the following types of categories are strict:

  • in posets,

  • in toposes,

  • in extensive categories,

  • in distributive categories,

  • in cartesian closed categories,

  • more generally, in any category where the products $\varnothing \times X$ exist and are initial (that is, where the product functor $- \times X$ preserves initial objects). In that case any map $f : X \to \varnothing$ is a section, hence an inverse, of the unique map $! : \varnothing \to X$, as shown by this diagram:

• Specifically the initial objects of Set, Cat, Top are all strict.

• John Baez showed on the Category Theory Zulip that the rational numbers are a strict initial object in the category of characteristic zero fields and ring homomorphisms. This implies that the rational numbers are a strict initial object in the category of ordered fields and ring homomorphisms.

• In the same discussion thread on the Category Theory Zulip Madeleine Birchfield showed that the integers are a strict initial object in the category of ordered integral domains and strictly monotonic ring homomorphisms.

At the other extreme, a zero object is a strict initial object only if the category is trivial (i.e. equivalent to the terminal category).

Related concepts

• An empty bundle is a morphism out of a strict initial object (an empty morphism), regarded as a bundle.

• formal dual concept: strict terminal object

• weakly initial object

References

• Aurelio Carboni, Stephen Lack, R. F. C. Walters, Def. 2.7 in: Introduction to extensive and distributive categories, JPAA 84 (1993) pp. 145-158 (doi:10.1016/0022-4049(93)90035-R)

Discussion as van Kampen colimits:

• Pawel Sobocinski, Tobias Heindel, Exp. 4.5 (i) in: Being Van Kampen is a universal property, Logical Methods in Computer Science, 7 1 (2011) [arXiv:1101.4594, doi:10.2168/LMCS-7(1:14)2011]