nLab
inhabited object

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Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

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Category Theory

Contents

Idea

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.

Definition

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

xX. \exists x \in X.

This is equivalent to saying that the unique map X1X \to 1 is an epimorphism. More generally, we can make the same definition in any regular category, in which case it is equivalent to X1X\to 1 being a regular epimorphism.

The term well-supported is also used for this notion (in general, the support of XX is the image of X1X\to 1).

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

On the other hand, XX is said to be externally or globally inhabited if there exists a morphism 1X1\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:Xx: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 11 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 “XX 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 1X1\to X (which is the same whether interpreted internally or externally), and the terms “global element” and “well-supported” are always unambiguous.

Examples

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

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

In a well-pointed topos

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 SetSet is a well-pointed topos whose terminal object 11 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 11 easily makes internal and external inhabitedness agree. We also present a constructive/intuitionistic proof of the following result about emptiness.

Proposition

In a well-pointed topos, an object XX 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 XU1X \to U \hookrightarrow 1 be the epi-mono factorization of the unique map X1X \to 1. Here m:U1m: U \hookrightarrow 1 is monic but not epic if X1X \to 1 is not epic, and we use this to prove U0U \cong 0. In that case, we have a map X0X \to 0, whence X0X \cong 0 since in a topos initial objects are strict. (For given X0X \to 0, the projection π 1:X×0X\pi_1: X \times 0 \to X has a section, and it follows that X0X \cong 0 since X×00X \times 0 \cong 0 and retracts of initial objects are initial. This also implies that for any VV the map i:0Vi: 0 \to V is monic: for it is obviously true that for any pair of maps h,k:A0h, k: A \to 0 we have that ih=iki h = i k implies h=kh = k, as AA is initial by strictness and this makes h=kh = k automatic.)

Let f:UΩf: U \to \Omega be the classifying map of the mono 1 U:UU1_U: U \to U, and let g:UΩg: U \to \Omega be the classifying map of the mono 0U0 \to U. There is no map 1U1 \to U, else m:U1m: U \to 1 would retract it and hence be epic. Hence it is vacuously true that fx=gxf x = g x for all x:1Ux: 1 \to U, and so f=gf = g by well-pointedness. Hence the subobjects 1 U1_U and 00 coincide, forcing U0U \cong 0. (In other words, the presence of a subobject classifier causes any generator to be a strong generator.)

Last revised on October 11, 2020 at 09:23:01. See the history of this page for a list of all contributions to it.