classifying topos for the theory of objects in nLab

Contents

Contents

Idea

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

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

This is because by a standard fact geometric morphisms

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

are equivalent to morphisms of sites

E←FinSet op E \leftarrow FinSet^{op}

hence to finite limit-preserving such functors. Since finite limits in FinSet opFinSet^{op} are finite colimits in FinSet and since FinSetFinSet 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 EE.

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

• write Fin∞GrpdFin\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 ∞(Fin∞Grpd op)PSh_\infty(Fin\infty Grpd^{op}) is the classifying (∞,1)-topos for objects;

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

Properties

Generic object

The generic or universal object UU is the inclusion FinSet↪SetFinSet\hookrightarrow Set: every object XX of EE arises from some geometric morphism ff as X≅f *(U)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]k[x] over a ground ring kk that ‘classifies’ elements of k-algebras AA via A≅Hom k(k[x],A)A\cong Hom_k(k[x], A), it is traditionally denoted 𝒮[U]\mathcal{S}[U], the ‘adjunction’ of the free (=generic) object to the base topos 𝒮\mathcal{S}, in our case SetSet.

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), 𝒮[U]\mathcal{S}[U] is shown to play the role of the real line R\mathbf R in functional analysis.

The relative case

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

An alternative characterization

The monoidal point of view

Since the category End f(Set)↪End(Set)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][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 opFinSet^{op} is the free cartesian monoidal category generated by an object (or by the terminal category), and its free cocompletion [FinSet,Set][FinSet, Set] is the free cartesian monoidally cocomplete category generated by an object. Thus [FinSet,Set][FinSet, Set] plays the role of a cartesian analogue to [ℙ op,Set][\mathbb{P}^{op}, Set], the free symmetric monoidally cocomplete category on an object, where ℙ=Core(FinSet)\mathbb{P} = Core(FinSet) is the core of FinSet, the permutation groupoid. And in the same way that [ℙ op,Set][\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][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→CatU:GrTop^{op}\to Cat from the opposite of the 2-category GrTopGrTop 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 𝒮[𝕆]\mathcal{S}[\mathbb{O}] is a representing object for UU.

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 ℰ\mathcal{E} to 𝒮[𝕆]\mathcal{S}[\mathbb{O}] are equivalent to the objects of ℰ\mathcal{E}, the sheaves over the generalized space of whatever ℰ\mathcal{E} classifies. But 𝒮[𝕆]\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 ℰ\mathcal{E} is sheaves over an ungeneralized space XX. The sheaf is equivalent to a local homeomorphism to XX, and the set-valued map takes points of XX to the stalks, the fibres of the local homeomorphism.

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

The object classifier and exponentiability

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

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

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 𝕆 *\mathbb{O}_\ast is, of course, classified by a topos, namely, the presheaf topos [FinSet *,Set][FinSet_\ast, Set] on the opposite FinSet * opFinSet_\ast^{op} of the category of finite pointed sets whose skeleton is Segal's category, hence [FinSet *,Set][FinSet_\ast, Set] is equivalent to the topos of “Γ\Gamma-sets”: cf. Gamma-space and for the role as a classifying space the following MO-discussion: link .³

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

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

GG is the inclusion functor FinSet→SetFinSet \to Set, mapping each natural number nn to its finite cardinal {0,…,n−1}\{0,\ldots,n-1\}.

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

• 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)