nLab
strict initial object
Contents
Context
Category theory
category theory

Concepts
Universal constructions
Theorems
Extensions
Applications
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
The empty set , among all sets, has two characteristic properties:

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

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

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

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:

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 ).

References
Last revised on November 29, 2022 at 14:05:54.
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