Contents

(0,1)-category

(0,1)-topos

# Contents

## Idea

In a poset $P$, a top of $P$ is a greatest element: an element $\top$ of $P$ such that $a \leq \top$ for every element $a$. Such a top may not exist; if it does, then it is unique.

In a proset, a top may be defined similarly, but it need not be unique. (However, it is still unique up the natural equivalence in the proset.)

A top of $P$ can also be understood as a meet of zero elements in $P$.

A poset that has both top and bottom is called bounded.

As a poset is a special kind of category, a top is simply a terminal object in that category.

The top of the poset of subsets or subobjects of a given set or object $A$ is always $A$ itself.

basic symbols used in logic

$\phantom{A}$symbol$\phantom{A}$$\phantom{A}$meaning$\phantom{A}$
$\phantom{A}$$\in$$\phantom{A}$element relation
$\phantom{A}$$\,:$$\phantom{A}$typing relation
$\phantom{A}$$=$$\phantom{A}$equality
$\phantom{A}$$\vdash$$\phantom{A}$$\phantom{A}$entailment / sequent$\phantom{A}$
$\phantom{A}$$\top$$\phantom{A}$$\phantom{A}$true / top$\phantom{A}$
$\phantom{A}$$\bot$$\phantom{A}$$\phantom{A}$false / bottom$\phantom{A}$
$\phantom{A}$$\Rightarrow$$\phantom{A}$implication
$\phantom{A}$$\Leftrightarrow$$\phantom{A}$logical equivalence
$\phantom{A}$$\not$$\phantom{A}$negation
$\phantom{A}$$\neq$$\phantom{A}$negation of equality / apartness$\phantom{A}$
$\phantom{A}$$\notin$$\phantom{A}$negation of element relation $\phantom{A}$
$\phantom{A}$$\not \not$$\phantom{A}$negation of negation$\phantom{A}$
$\phantom{A}$$\exists$$\phantom{A}$existential quantification$\phantom{A}$
$\phantom{A}$$\forall$$\phantom{A}$universal quantification$\phantom{A}$
$\phantom{A}$$\wedge$$\phantom{A}$logical conjunction
$\phantom{A}$$\vee$$\phantom{A}$logical disjunction
$\phantom{A}$$\otimes$$\phantom{A}$$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$$\phantom{A}$multiplicative disjunction$\phantom{A}$

Last revised on July 3, 2018 at 03:04:45. See the history of this page for a list of all contributions to it.