nLab strict initial object



Category theory

Limits and colimits



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:


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

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


  • The Cartesian product of any object XX with a strict initial object is isomorphic to the strict initial object, X×X \times \varnothing \simeq \varnothing, because the projection pr 1:×Xpr_1 \colon \varnothing \times X \to \varnothing exists by definition of Cartesian products, whence (1) implies that it is an isomorphism

    ×Xpr 1. \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/C/\varnothing be equivalent to the terminal category 1\mathbf{1}.


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).


Discussion as van Kampen colimits:

Last revised on March 8, 2024 at 13:02:15. See the history of this page for a list of all contributions to it.