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, van Kampenness 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.

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

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]