nLab
strict terminal object
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Category theory
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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

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end and coend

fibered limit

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
The trivial ring , among all unital rings, has two characteristic properties:

there is a unique function into the trivial ring, from any other unital ring;

there is no function from the trivial ring, except to itself.

The first property generalizes to arbitrary categories as the property of a terminal object .

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

Definition
A terminal object $\mathbf{1}$ is called a strict terminal object if every morphism from $\mathbf{1}$ is an isomorphism :

$\mathbf{1}
\overset{\simeq}{\longrightarrow}
\X
\,.$

In other words, a strict terminal object is a maximal terminal object .

Examples
Trivial unital ring

Trivial Boolean algebra

Trivial absorption monoid

In general, for any algebraic theory with two constants $0$ and $1$ and a binary operation for which $0$ is a (left) absorbing element and $1$ is a (left) unit, the trivial model is strictly terminal.

References
Last revised on March 17, 2023 at 15:01:08.
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