nLab Jónsson-Tarski topos



Topos Theory

topos theory



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The Jónsson-Tarski topos is ‘the topos analogue of the Cantor space’.1


The Jónsson-Tarski topos 𝒥 2\mathcal{J}_2 is the category of Jónsson-Tarski algebras considered as topos, i.e. its objects are sets XX together with an isomorphism XX×XX\to X\times X and morphisms are functions that commute with the structure isomorphisms.


  • 𝒥 2\mathcal{J}_2 is an example of an variety of algebras that is also a topos (cf. Johnstone 1985).

  • 𝒥 2=Sh(M 2,J)\mathcal{J}_2=Sh(M_2,J) where M 2M_2 is the free monoid on two generators a,ba, b and JJ is the coverage whose only covering family on the unique object \cdot of M 2M_2 is {a:,b:}\{a:\cdot\rightarrow \cdot ,b:\cdot\rightarrow\cdot\}. So 𝒥 2\mathcal{J}_2 is in a fact a Grothendieck topos.

  • M 2M_2, as a free monoid, is cancellative, hence all-monic and, accordingly, 𝒥 2\mathcal{J}_2 is an étendue (Peter Freyd). It is discussed from this perspective as a petit topos for labeled graphs in (Lawvere 1989).

  • Actually, Freyd observed that 𝒥 2/F(1)Sh(2 )\mathcal{J}_2/F(1)\cong Sh(2^\mathbb{N}) with F(1)F(1) the free Jónsson-Tarski algebra on one generator and 2 2^\mathbb{N} the Cantor space - this motivates the above quote from Bunge&Funk: 𝒥 2\mathcal{J}_2 looks locally like (the sheaf topos on) 2 2^\mathbb{N} ! Sh(2 )Sh(2^\mathbb{N}) classifies ‘ subsets of \mathbb{N} ‘ in the sense that for cocomplete \mathcal{E} geometric morphisms Sh(2 )\mathcal{E}\to Sh(2^\mathbb{N}) correspond to morphisms Δ()Δ(2)\Delta(\mathbb{N})\to\Delta(2) in \mathcal{E}, with Δ\Delta the constant sheaf functor? (cf. Mac Lane& Moerdijk, ex.VIII.10, pp.470-71, ex.X.6, pp.572-73).


The idea to consider generalizations of 𝒥 2\mathcal{J}_2 seemed to have appeared first in the context of work on étendues (Rosenthal 1981, Lawvere 1989).

Rosenthal’s approach

K. Rosenthal (1981) starts from two basic facts about étendues, namely that Set 𝒞 opSet^{\mathcal{C}^{op}} is an étendue iff all morphisms in 𝒞\mathcal{C} are monic, and that Sh(,J)Sh(\mathcal{E},J) is an étendue if \mathcal{E} is an étendue. His goal is to construct étendues from an all-monic 𝒞\mathcal{C} by defining a topology JJ on Set 𝒞 opSet^{\mathcal{C}^{op}} from a functor H:𝒞SetH:\mathcal{C}\to Set satisfying:

  1. H(f)H(f) is monic ,and

  2. if xIm(H(f))Im(H(g))x\in Im(H(f))\cap Im(H(g)) then there is k𝒞k\in\mathcal{C} with kfk\leq f and kgk\leq g such that xIm(H(k))x\in Im(H(k)).

Now given X𝒞X\in\mathcal{C} and a sieve BΩ(X)B\in\Omega(X) define a sieve

j X(B):={f𝒞|cod(f)=XIm(H(f)) gBIm(H(g))}.j_X(B):=\{\quad f\in\mathcal{C}| cod(f)=X\quad\wedge\quad Im(H(f))\subseteq\bigcup _{g\in B} Im(H(g))\quad\}.

The resulting map j:ΩΩj:\Omega\to\Omega is a topology.

For 𝒞 op= 2\mathcal{C}^{op}=\mathcal{M}_2, the free monoid on two generators, and H(X)=2 H(X)=2^\mathbb{N}, the functor constantly the Cantor set, this yields 𝒥 2\mathcal{J}_2.

The generalization to 𝒞 op= \mathcal{C}^{op}=\mathcal{M}_\infty, the free monoid on countably infinite many generators, and the Baire space 𝔹= \mathbb{B}=\mathbb{N}^\mathbb{N} exhibits the infinite Jónsson-Tarski topos 𝒥 \mathcal{J}_\infty, i.e. the category of sets AA with an isomorphism to A A^\mathbb{N}, as Sh( )Sh(\mathbb{N}^\mathbb{N}) locally.

Jónsson-Tarski toposes via profunctors

Work on a categorical concept of self-similarity led T. Leinster (2007) to another generalization of the Jónsson-Tarski topos. He starts from the observation that any profunctor M:𝒜M:\mathcal{A} ⇸\mathcal{B} comes with an adjunction M[M,]:Set 𝒜 opSet op-\otimes M\dashv [M,-]:Set^{\mathcal{A}^{op}}\to Set^{\mathcal{B}^{op}} where the left adjoint stems from profunctor composition and the right adjoint is defined for presheaves XSet opX\in Set^{\mathcal{B}^{op}} and a𝒜a\in \mathcal{A} as [M,X](a):=Nat(M(,a),X)[ M,X ] (a) := Nat(M(-,a),X) i.e. the set of natural transformations from M(,a)M(-,a) to XX or, in other words, Hom(M(,a),X)Hom(M(-,a),X) in Set opSet^{\mathcal{B}^{op}}.

For an endoprofunctor M:𝒜𝒜M:\mathcal{A} ⇸\mathcal{A} define a Jónsson-Tarski M-algebra as a pair (X,ξ)(X,\xi) where XSet 𝒜 opX\in Set^{\mathcal{A}^{op}} and ξ\xi a natural isomorphism X[M,X]X\overset{\simeq}{\to} [M,X]. The resulting category 𝒥 M\mathcal{J}_M is a topos since a site (𝒜 M,J)(\mathcal{A}_M,J) can be constructed from 𝒜\mathcal{A} by adjoining new arrows bmab\overset{m}{\to} a for each b,a𝒜b,a\in\mathcal{A} and mM(b,a)m\in M(b,a) with covers J(a)J(a) the set of these arrows (cf. Worrell 2002, Leinster 2007). Furthermore, 𝒥 M\mathcal{J}_M is monadic over Set 𝒜 opSet^{\mathcal{A}^{op}}.

The classical Jónsson-Tarski topos 𝒥 2\mathcal{J}_2 arises from this process by taking M=2:11M=\mathbf{2}:\mathbf{1} ⇸\mathbf{1} assigning to the unique object (,)(\bullet,\bullet) of the category 1×1\mathbf{1}\times\mathbf{1} the two element set {,*}\{\star,\ast\}. A presheaf on 1\mathbf{1} is just a set XX whence [2,X]()[\mathbf{2}, X](\bullet) is just the set of all maps {,*}X\{\star,\ast\}\to X i. e. X×XX\times X. The site for it is the free category generated by the graph with one node \bullet and two edges ,*\star,\ast i. e. the free monoid on two generators and with coverage {,*}\{\star,\ast\}.

Some further properties of 𝒥 2\mathcal{J}_2 are discussed in the answer to this


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

  • Peter Johnstone, When is a Variety a Topos? , Algebra Universalis 21 (1985) pp.198-212.

  • Peter Johnstone, Collapsed Toposes as Bitopological Spaces , pp.19-35 in Categorical Topology , World Scientific Singapore 1989.

  • Peter Johnstone, Collapsed Toposes and Cartesian Closed Varieties , JA 129 (1990) pp.446-480.

  • Peter Johnstone, Sketches of an Elephant vol. I , Oxford UP 2002. (sec. A2.1, p.80)

  • P. T. Johnstone, A. J. Power, T. Tsujishita, H. Watanabe, J. Worrell, The structure of categories of coalgebras , Theoret. Comp. Sci 260 (2001) pp.87-117.

  • F. William Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs , Cont. Math. 92 (1989) pp.261-299.

  • Tom Leinster, Jónsson-Tarski toposes, Talk Nice 2007. (slides)

  • S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (pp.470-471)

  • Kimmo I. Rosenthal, Étendues and Categories with Monic Maps , JPAA 22 (1981) pp.193-212.

  • James Worrell, A Note on Coalgebras and Presheaves , Electronic Notes in Theoretical Computer Science 65 no.3 (2003) pp.1-10.

  1. Quote from Bunge&Funk (2006, p.183).

Last revised on May 4, 2022 at 10:03:55. See the history of this page for a list of all contributions to it.