In category theory and specifically in topos theory, the notion of *inhabited object* in a general topos is a generalization of the notion of inhabited set from the archetypal topos Set. It is the semantics of an inhabited type in type theory.

An object $X$ in a topos $\mathcal{T}$ is said to be **internally inhabited** if in the internal logic of the topos it is true that

$\exists x \in X.$

This is equivalent to saying that the unique map $X \to 1$ is an epimorphism. More generally, we can make the same definition in any regular category, in which case it is equivalent to $X\to 1$ being a regular epimorphism.

The term **well-supported** is also used for this notion (in general, the support object of $X$ is the image of $X\to 1$).

In terms of (∞,1)-category theory, internally inhabited means *(-1)-connected*.

On the other hand, $X$ is said to be **externally** or **globally inhabited** if there exists a morphism $1\to X$, i.e. a global element. The adjective “externally” is perhaps misleading, since this notion can also be expressed in the internal type theory of the topos (as having an element/term $x:X$), though not in its more usual internal higher-order logic. The internal assertion “every internally inhabited object is globally inhabited” is a version of the propositional axiom of choice (PAC).

Every globally inhabited object is internally inhabited, since every split epimorphism is a regular epimorphism. The converse is true if $1$ is projective, as is the case in a well-pointed topos (such as Set). (This may seem to contradict the above, since PAC can fail even in (constructively) well-pointed toposes, but the point is that PAC also says something about *families* of inhabited objects rather than single “globally defined” objects.)

Some sources use “$X$ is inhabited” in the global sense, while others use the term “inhabited” only internally. Regardless, a pointed object always means one *equipped with* a global element $1\to X$ (which is the same whether interpreted internally or externally), and the terms “global element” and “well-supported” are always unambiguous.

For sheaf toposes with enough points, in which epimorphisms are stalkwise epimorphisms, an object is (internally) inhabited if the germ of $X$ over every stalk is an inhabited set.

One situation where this plays a role is in the study of certain smooth toposes with objects $\mathbb{I}$ of *invertible infinitesimals* . There is an immediate definition of such a topos, the topos called $\mathcal{Z}$ at Models for Smooth Infinitesimal Analysis, for which this object exists, but is not inhabited. Only the weaker internal statement $\not \not \exists x \in \mathbb{I}$ is true.

But for some useful constructions in these toposes, such as for giving an internal definition of distributions as genuine functions (internally), it is desirable to have $\mathbb{I}$ be inhabited. In the above situation this is achieved by forcing the existence of invertible infinitesimal elements. The result is the refined topos denoted $\mathcal{B}$ at Models for Smooth Infinitesimal Analysis.

The category Set is a very special sort of topos. A generally useful principle in formulating 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.

As remarked above, projectivity of $1$ easily makes internal and external inhabitedness agree. We also present a constructive/intuitionistic proof of the following result about *emptiness*.

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.

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.)

Last revised on March 10, 2023 at 04:57:17. See the history of this page for a list of all contributions to it.