nLab classifying topos for the theory of objects

Idea

The classifying topos $\mathcal{S}[\mathbb{O}]$ for the theory of objects $\mathbb{O}$, or the object classifier, as it is also called¹, is the presheaf topos $[FinSet, Set]$ on the opposite category of FinSet.

What motivates the terminology, is that for any topos $E$, geometric morphisms $E \to \mathcal{S}[\mathbb{O}]$ correspond to objects of $E$.

This is because by a standard fact geometric morphisms

$$E \to PSh(FinSet^{op})$$

are equivalent to morphisms of sites

$$E \leftarrow FinSet^{op}$$

hence to finite limit-preserving such functors. Since finite limits in $FinSet^{op}$ are finite colimits in FinSet and since $FinSet$ is generated under finite colimits from the singleton set $\ast$, such functors are uniquely determined by their image of $\ast$, hence by a choice of object in $E$.

The construction is due to Gavin Wraith and constituted an important step towards the general theorem on the existence of classifying toposes for geometric theories in the early development of topos theory.

Similarly

• $PSh(FinSet_\ast^{op})$ is the classifying topos for pointed objects.

• write $Fin\infty Grpd$ for the full sub-(∞,1)-category on ∞Grpd which is generated under finite (∞,1)-colimits from the point $\ast$ (HA, def. 1.4.2.8), then the (∞,1)-presheaf (∞,1)-topos $PSh_\infty(Fin\infty Grpd^{op})$ is the classifying (∞,1)-topos for objects;

• write $Fin\infty Grpd_\ast$ for pointed finite $\infty$-groupoids in this sense, then $PSh_\infty((Fin\infty Grpd_\ast)^{op})$ is the classifying $(\infty,1)$-topos for pointed objects. See also at spectrum object via excisive functors.

Properties

Generic object

The generic or universal object $U$ is the inclusion $FinSet\hookrightarrow Set$: every object $X$ of $E$ arises from some geometric morphism $f$ as $X\cong f^\ast(U)$.²

The symbolic link to algebra

As the role of the object classifier bears some resemblance to the role of the polynomial ring $k[x]$ over a ground ring $k$ that ‘classifies’ elements of k-algebras $A$ via $A\cong Hom_k(k[x], A)$, it is traditionally denoted $\mathcal{S}[U]$, the ‘adjunction’ of the free (=generic) object to the base topos $\mathcal{S}$, in our case $Set$.

The analogy between topos theory and algebra is pursued further in Bunge&Funk (2006) where, in the context of topos distributions and the ‘symmetric algebra’ of a topos (aka the symmetric topos), $\mathcal{S}[U]$ is shown to play the role of the real line $\mathbf R$ in functional analysis.

The relative case

What concerns base toposes $\mathcal{S}$ other than $Set$, it is a theorem due to Andreas Blass (Blass 1989) that $\mathcal{S}$ has an object classifier $\mathcal{S}[\mathbb{O}]$ precisely if $\mathcal{S}$ has a natural number object.

A consequence of this, discussed in sec. B4.2 of (Johnstone 2002,I p.431), is that classifying toposes for geometric theories over $\mathcal{S}$ exist precisely if the object classifier $\mathcal{S}[\mathbb{O}]$ exists.

An alternative characterization

Proposition

The classifying topos $[FinSet, Set]$ is equivalent to the category of finitary endofunctors on Set (those that commute with filtered colimits):

$$[FinSet, Set] \simeq End_f(Set) \,.$$

Proof

Because every set is the filtered colimit over its finite subsets.

Constructively we should take a little care over what is meant by “finite set”. A set is the filtered colimit of its Kuratowski finite subsets, but a Kuratowski finite set is the image of some finite cardinal $\{0,\ldots, n-1\}$. The issue is that, unless the superset has decidable equality, we are not necessarily able to eliminate duplicates from the list of elements in the Kuratowski finite subset. We find that the set is still a filtered colimit of finite cardinals, though not necessarily subsets. The category $FinSet$ can be taken to have the natural numbers as its objects, with morphisms being functions between the corresponding finite cardinals.

The monoidal point of view

Since the category $End_f(Set) \hookrightarrow End(Set)$ of finitary functors? is naturally a monoidal category under composition, this induces the structure of a (non-cartesian) monoidal category also on the classifying topos $[FinSet, Set]$ and hence makes it a monoidal topos. A monoid with respect to this monoidal structure is equivalently a finitary monad.

A related point of view is that $FinSet^{op}$ is the free cartesian monoidal category generated by an object (or by the terminal category), and its free cocompletion $[FinSet, Set]$ is the free cartesian monoidally cocomplete category generated by an object. Thus $[FinSet, Set]$ plays the role of a cartesian analogue to $[\mathbb{P}^{op}, Set]$, the free symmetric monoidally cocomplete category on an object, where $\mathbb{P} = Core(FinSet)$ is the core of FinSet, the permutation groupoid. And in the same way that $[\mathbb{P}^{op}, Set]$ is a monoidal topos whose monoids are symmetric or permutative operads (as discussed at operad – a detailed conceptual treatment), so $[FinSet, Set]$ is seen as a monoidal topos whose monoids are cartesian operads, aka Lawvere theories. Some material on this can be found at Towards a doctrine of operads.

The functorial point of view

Consider the forgetful 2-functor $U:GrTop^{op}\to Cat$ from the opposite of the 2-category $GrTop$ of Grothendieck toposes and geometric morphisms that maps a topos to its underlying category and a geometric morphism to its inverse image functor. The object classifier $\mathcal{S}[\mathbb{O}]$ is a representing object for $U$.

For more on this functorial approach to geometric theories see at geometric theory#FunctorialDefinition or Johnstone (2002, vol.1 pp.424ff).

The generalized space point of view

The universal property says that maps (geometric morphisms) from a Grothendieck topos $\mathcal{E}$ to $\mathcal{S}[\mathbb{O}]$ are equivalent to the objects of $\mathcal{E}$, the sheaves over the generalized space of whatever $\mathcal{E}$ classifies. But $\mathcal{S}[\mathbb{O}]$ is the generalized space of “sets”, so sheaves should be understood as continuous set-valued maps. This is seen most clearly when $\mathcal{E}$ is sheaves over an ungeneralized space $X$. The sheaf is equivalent to a local homeomorphism to $X$, and the set-valued map takes points of $X$ to the stalks, the fibres of the local homeomorphism.

So what are the sheaves over $\mathcal{S}[\mathbb{O}]$, i.e. its objects? They are continuous maps $F$ from the space of sets to itself. All continuous maps (geometric morphisms) preserve filtered colimits of points, and it follows that $F$ is determined by its action on finite cardinals. Hence $\mathcal{S}[\mathbb{O}]$ is - at least - a subcategory of $[FinSet, Set]$.

The object classifier and exponentiability

Recall that a locale $X$ is called exponentiable (in the category of locales) if the exponential $Y^X$ exists for all locales $Y$. Interestingly, exponentiability of $X$ hinges on the existence of the single exponential $S^X$ where $S$ is the Sierpinski space: $Y^X$ exists for all $Y$ iff $S^X$ exists.

In the 2-category $GrTop$ the object classifier $\mathcal{S}[\mathbb{O}]$ takes over the role of the Sierpinski space and we have the following

Proposition

A Grothendieck topos $\mathcal{E}$ is exponentiable iff the exponential $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ exists.

This result is due to Johnstone-Joyal (1982, p.257) and occurs as theorem 4.3.1 of Johnstone (2002, vol.1 p.433).

The pointed object classifier

Pointed objects are important in homotopy theory. Similarly to objects, they arise as models of a geometric theory and this theory of pointed objects $\mathbb{O}_\ast$ is, of course, classified by a topos, namely, the presheaf topos $[FinSet_\ast, Set]$ on the opposite $FinSet_\ast^{op}$ of the category of finite pointed sets whose skeleton is Segal's category, hence $[FinSet_\ast, Set]$ is equivalent to the topos of “$\Gamma$-sets”: cf. Gamma-space and for the role as a classifying space the following MO-discussion: link .³

Since the theory $\mathbb{O}_\ast$ is an extension of $\mathbb{O}$, it can be treated relatively as an internal theory in $\mathcal{S}[\mathbb{O}]$. There it is just the theory of elements of the generic set $G$, in other words the propositional theory for the discrete space of $G$. Discrete spaces are always got as slice toposes, so $\mathcal{S}[\mathbb{O}_\ast]$ is equivalent to

$$[FinSet, Set]/G\simeq [(\underset{FinSet^{op}}{\int} G)^{op},Set]\;.$$

$G$ is the inclusion functor $FinSet \to Set$, mapping each natural number $n$ to its finite cardinal $\{0,\ldots,n-1\}$.

The forgetful map (geometric morphism) from $\mathcal{S}[\mathbb{O}_\ast]$ to $\mathcal{S}[\mathbb{O}]$, forgetting the point, is the generic local homeomorphism. Every local homeomorphism is a bipullback of it.

Related entries

• theory of objects

• classifying topos

• geometric theory

• exponentiable topos

• Sierpinski space

References

• P. T. Johnstone, Topos Theory , Academic Press New York 1977 (Dover reprint 2014). (sec. 6.3)

• Peter Johnstone, Sketches of an Elephant , Oxford UP 2002. (sections B4.2 pp.424-432, D3.2 pp.901-910)

• Peter Johnstone, André Joyal, Continuous categories and exponentiable toposes , JPAA 25 (1982) pp.255-296.

• Peter Johnstone, Gavin Wraith, Algebraic Theories in a Topos , pp.141-242 in LNM 661 Springer Heidelberg 1978. (ch. IV, pp.175ff)

• Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and Logic, Springer Heidelberg 1994. (sec.VIII.4)

• Andreas Blass, Classifying topoi and the axiom of infinity , Algebra Universalis 26 (1989) 341-345 [doi:10.1007/BF01211840]

• Andreas Blass, Andrej Scedrov, Classifying topoi and finite forcing, Journal of Pure and Applied Algebra 28 (1983) 111-140 [doi;10.1016/0022-4049(83)90085-3]

• Marta Bunge, Jonathon Funk, Singular Coverings of Toposes , Springer LNM 1890 Heidelberg 2006.

• Richard Garner, Lawvere theories, finitary monads and Cauchy-completion , JPAA 218 no.11 (2014) pp.1973–1988. (preprint)

• Gavin Wraith, Topos Elémentaire et Arithmétique , Cah. Top. Géom. Diff. Cat. XIV-2 (1973) pp.217-218. (proceedings colloque Amiens 1973, 3.92 MB)

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1. As this is not to be confused with the notion of an object classifier in an (∞,1)-topos, we prefer to call it in full the classifying topos for the theory of objects. ↩

2. For another remarkable property of this inclusion functor see at ultrafilter monad. ↩

3. More generally, classifying toposes for universal Horn theories $\mathbb{T}$ correspond to the respective toposes of covariant set-valued functors on the category of finitely presentable models of $\mathbb{T}$ (Blass & Scedrov (1983)). ↩