nLab
strict initial object
Context
Category theory
category theory
Concepts
Universal constructions
Theorems
Extensions
Applications
Limits and colimits
limits and colimits
1Categorical

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
2Categorical
(∞,1)Categorical
Modelcategorical
Contents
Definition
An initial object $\emptyset$ is called a strict initial object if any morphism $x\to \emptyset$ must be an isomorphism.
Examples
The initial objects of a poset, of Set, Cat, Top, and of any topos (more generally of any extensive category) are strict.
At the other extreme, a zero object is only a strict initial object if the category is trivial (equivalent to the terminal category).