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Idea

In set theory, an inhabited set or occupied set is a set that contains an element.

More generally, in type theory an inhabited type is a type that has a term.

At least assuming classical logic, this is the same thing as a set that is not empty. Usually inhabited sets are simply called ‘non-empty’, but the positive word ‘inhabited’ reminds us that this is the simpler notion, which emptiness is defined as the negation of.

The terms ‘inhabited’ and ‘occupied’ come from constructive mathematics. In constructive mathematics (such as the internal logic of some topos or generally in type theory), a set/type that is not empty is not already necessarily inhabited. This is because double negation is nontrivial in intuitionistic logic. All the same, many constructive mathematicians use the old word ‘non-empty’ with the understanding that it really means inhabited.

There is a distinction between ‘inhabited’ and ‘occupied’ spaces in Abstract Stone Duality (which probably corresponds to something about locales, should explain that here).

An inhabited set is the special case of an inhabited object in the topos Set.

Internal and external notions

In topos theory, one should distinguish two notions of “inhabited”. An object $X$ is internally inhabited if the unique morphism $X \to 1$ to a terminal object is an epimorphism. It is externally inhabited if there exists a global element, i.e., a morphism $1 \to X$; such forces $X \to 1$ to be a retraction and hence epic, i.e., forces $X$ to be internally inhabited. The two notions coincide if the terminal object is projective.

A generally useful principle in formulating appropriate categorical properties of Set is that external and internal notions should coincide. A general default understanding across the nLab is that $Set$ is a well-pointed topos whose terminal object $1$ is projective and indecomposable. These latter properties of the terminal object hold for any well-pointed topos if the metalogic is classical; for purposes of constructive mathematics, we usually agree to add them in. In any event, we present a constructive/intuitionistic proof of the following result.

Proposition

In a well-pointed topos, an object $X$ which is not internally inhabited is empty (is an initial object).

Note that the negation “not” in this statement is an external one, which is what makes it nontrivial. If an object is internally “not inhabited”, then it is empty essentially by definition.

Proof

Let $X \to U \hookrightarrow 1$ be the epi-mono factorization of the unique map $X \to 1$. Here $m: U \hookrightarrow 1$ is monic but not epic if $X \to 1$ is not epic, and we use this to prove $U \cong 0$. In that case, we have a map $X \to 0$, whence $X \cong 0$ since in a topos initial objects are strict. (For given $X \to 0$, the projection $\pi_1: X \times 0 \to X$ has a section, and it follows that $X \cong 0$ since $X \times 0 \cong 0$ and retracts of initial objects are initial. This also implies that for any $V$ the map $i: 0 \to V$ is monic: for it is obviously true that for any pair of maps $h, k: A \to 0$ we have that $i h = i k$ implies $h = k$, as $A$ is initial by strictness and this makes $h = k$ automatic.)

Let $f: U \to \Omega$ be the classifying map of the mono $1_U: U \to U$, and let $g: U \to \Omega$ be the classifying map of the mono $0 \to U$. There is no map $1 \to U$, else $m: U \to 1$ would retract it and hence be epic. Hence it is vacuously true that $f x = g x$ for all $x: 1 \to U$, and so $f = g$ by well-pointedness. Hence the subobjects $1_U$ and $0$ coincide, forcing $U \cong 0$. (In other words, the presence of a subobject classifier causes any generator to be a strong generator.)

Revised on May 19, 2014 06:15:37 by Mike Shulman (192.195.154.58)