Discrete objects

Examples

# Discrete objects

## Idea

A discrete space is, in general, an object of a concrete category $Sp$ of spaces that is free on its own underlying set. More generally, the notion can be applied relative to any forgetful functor.

Note: This page is about the “cohesive” or “topological” notion of discreteness. In 2-category theory the term “discrete object” is also often used for 0-truncated objects. For this usage, see discrete morphism instead.

## Definition

A discrete space must, in particular, be a free object for the forgetful functor $U\colon Sp\to Set$, i.e. in the image of its left adjoint $F: Set \to Sp$. However, this is not sufficient for it to be free on its own underlying set; we must also require that the counit $F U X\to X$ be an isomorphism.

Thus, we say that $U\colon Sp \to Set$ (or more generally, any functor) has discrete spaces or discrete objects if it has a fully faithful left adjoint. This ensures that the functor

$Set \stackrel{F}\to Sp \stackrel{U}\to Set$

is (naturally isomorphic to) the identity functor on Set. This is true, for example, if $Sp$ is Top, Diff, Loc, etc.

Assuming that $U$ is faithful (as it is when $Sp$ is a concrete category), we can characterise a discrete space $X$ as one such that every function from $X$ to $Y$ (for $Y$ any space) is a morphism of spaces. (More precisely, this means that every function from $U(X)$ to $U(Y)$ is the image under $U$ of a morphism from $X$ to $Y$.)

The dual notion is a codiscrete object.

## Examples

### Discrete geometric spaces

The best known example is a discrete topological space, that is one, $X$, in which all subsets of $X$ are open in the topology. This is the discrete topology on $X$. If $X$ is discrete in this sense, then its diagonal map $\Delta: X \to X \times X$ is open. The converse also holds: if the diagonal $\Delta(X)$ is open, then so is $i_x^{-1}(\Delta(X)) = \{x\}$ for any $x \in X$, where $i_x(y) \coloneqq (x, y)$.

This same space serves as a discrete object in many subcategories and supercategories of $Top$, from convergence spaces (where the only proper filter that converges to a point is the free ultrafilter at that point) to (say) paracompact Hausdorff spaces or manifolds (because a discrete topological space has those properties).

It is also sober and thus serves as a discrete locale, whose corresponding frame is the power set of $X$; see CABA. (Note that Loc is not concrete over Set.). A locale is discrete if and only if $X \to X \times X$ is open and $X \to 1$ is also open. A locale that satisfies the latter condition is called overt; note that every locale is $T_0$ while every topological space is overt. Moreover, in classical mathematics, every locale is overt, but the notion is important when internalizing in toposes.

A discrete uniform space $X$ has all reflexive relations as entourages, or equivalently all covers as uniform covers. It is the only uniformity (on a given set) whose underlying topology is discrete.

Strictly speaking, there is no discrete metric space on any set with more than one element, because the forgetful functor has no left adjoint. However, there is a discrete extended metric space, given by $d(x,y) = \infty$ whenever $x \ne y$. More usually, the term ‘discrete metric’ is used when $d(x,y) = 1$ for $x \ne y$, which is discrete in the category of metric spaces of diameter at most $1$. (Comparing the adjoint functor theorem, the problem with $Met$ is that it generally lacks infinitary products; in contrast, $Ext Met$ and $Met_1$ are complete.)

In Abstract Stone Duality, a space is called discrete if the diagonal map $\delta: X \to X \times X$ is open, which corresponds to the existence of an equality relation on $X$; discrete spaces as described above correspond to discrete overt spaces in ASD.

### Local toposes

Any local topos has discrete and codiscrete objects. By definition, a local topos $\mathbf{H}$ comes with an adjoint triple of functors

$\mathbf{H} \stackrel{\overset{Disc}{\hookleftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}} \mathbf{B}$

to a base topos $\mathbf{B}$ (for instance Set), for which both $Disc$ and $Codisc$ are fully faithful. Thus, a discrete object is one in the essential image of the functor $Disc$. Note that $\Gamma$ is not generally faithful in this case.

Even more generally, $\mathbf{H}$ may be a local (∞,1)-topos. For more on the discrete objects in such a context see discrete ∞-groupoid .

$(\flat \dashv \sharp) \coloneqq ( Disc \Gamma \dashv coDisc \Gamma) \,,$

the flat modality and the sharp modality. The discrete objects are precisely the modal types for the flat modality. The codiscrete objects are the modal types for the sharp modality.

### Topological categories, fibrations, and final lifts

Every topological concrete category has discrete (and also codiscrete) spaces

More generally, if $U$ is an opfibration and $Sp$ has an initial object preserved by $U$, then $Sp$ has discrete objects: the discrete object on $X$ can be obtained as $i_!(0)$ where $0$ is the initial object of $Sp$ and $i\colon \emptyset \to X$ is the unique map from the initial object in $Set$ (or whatever underlying category). (Conversely, if $Sp$ has discrete objects and pushouts preserved by $U$, then $U$ is an opfibration.)

Discrete objects can also be characterized as final lifts for empty sinks.

### In simplicial sets

The category sSet of simplicial sets is a local topos (in fact a cohesive topos).

### Discrete cellular/categorical structures

Often one calls a cellular structure, such as those appearing in higher category theory, discrete if it is in the essential image of the inclusion of Set.

For instance, one may speak of a discrete category as a category that is equivalent (or, in some cases, isomorphic) to one which has only identity morphisms. This concept has a generalization to a notion of discrete object in a 2-category.

An alternative terminology for this use of “discrete” is 0-truncated, or more precisely (0,0)-truncated. A discrete groupoid in this sense is a homotopy 0-type, or simply a 0-type. This terminology may be preferable to “discrete” in this context, notably when one is dealing with higher categorical structures that are in addition equipped with geometric structure. For instance, when dealing with a topological category there is otherwise ambiguity in what it means to say that it is “discrete”: it could either mean that its underlying topological spaces (of objects and of morphisms) are discrete spaces, or it could mean that it has no nontrivial morphisms, but possibly a non-discrete topological space of objects.

###### Remark

In some cases, the cellular notion of “discreteness” for higher categories can be seen as a special case of the spatial notion of discreteness — often the 1-category of shapes will have a functor to sets for which the cellularly discrete objects are the discrete objects in the sense considered on this page. For instance, this is the case for simplicial sets, which form a local topos over Set. The discrete objects relative to this notion of cohesion are precisely the simplicial sets that are constant on a given ordinary set, hence those that are “discrete” in the cellular sense.

### In $\infty$-toposes

The definition of discrete objects has the evident generalization from category theory to (∞,1)-category theory/homotopy theory. One noteworthy aspect of discrete objects in the context of homotopy theory is that there they are intimately related to the notion of cohomology.

For $\mathcal{X}$ an (∞,1)-sheaf (∞,1)-topos with global section geometric morphism

$(\Delta \dashv \Gamma) \colon \mathcal{X} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd$

then for $X \in \mathcal{X}$ any object and $A \in$ ∞Grpd any object, one says that

$H(X,A) \coloneqq \pi_0 \mathcal{X}(X,\Delta(A)) \in Set$

is the cohomology of $X$ with (locally) constant coefficients in $A$. (Here on the right $\mathcal{X}(-,-)$ denotes the (∞,1)-categorical hom-space.)

Now if $\mathcal{X}$ has discrete objects in the sense that $\Delta \colon \infty Grpd \to \mathcal{X}$ is a full and faithful (∞,1)-functor, then it follows immediately from the definitions that the cohomology of discrete objects with constant coefficients in $\mathcal{X}$ equals the cohomology in ∞Grpd, which is standard “nonabelian cohomology”:

$(\mathcal{X} \, has\, discrete\, objects) \Leftrightarrow ( \forall_{S,A \in \infty Grpd} \mathcal{X}(\Delta S, \Delta A) \simeq \infty Grpd(S,A) ) \,.$

Conversely: the failure of the cohomology with constant coefficients of objects in the image of $\Delta$ to coincide with standard cohomology is a measure for $\Delta$ not respecting discrete objects.

For example the natural numbers object $\mathbf{N} \simeq \Delta \mathbb{N}$ in the (∞,1)-sheaf (∞,1)-topos over some topological spaces fails to be a discrete object. Accordingly in this case the natural numbers object can have nontrivial higher cohomology with constant coefficients, see for instance (Blass 83, Shulman 13).

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$