nLab Cole's theory of spectrum



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In the mid 1970s Julian Cole proposed a topos-theoretic construction of spectra in geometry arising in the sense of spectrum of a commutative ring but for more general (algebraic) theories, as right adjoints to forgetful functors that generalized M. Hakim's approach to locally ringed toposes.

Basic ingredients are pairs of geometric theories 𝕊\mathbb{S} and 𝕋\mathbb{T} over the same language such that 𝕋\mathbb{T} results from 𝕊\mathbb{S} by addition of further axioms. Then 𝕋Mod \mathbb{T}-Mod_\mathcal{E} is a full subcategory of 𝕊Mod \mathbb{S}-Mod_\mathcal{E} and the spectrum construction can be viewed as a sort of generalization of a right adjoint to the inclusion. The quotient relation between the two theories gives the construction a model-theoretic flavor.


Let 𝕋\mathbb{T} be geometric theory. The 2-category 𝕋𝔗𝔬𝔭\mathbb{T}-\mathfrak{Top} of T-modelled toposes is given as follows:

  • objects are pairs (,M)(\mathcal{E},M) where \mathcal{E} is a topos and MM a 𝕋\mathbb{T}-model in \mathcal{E},

  • 1-cells (,L)(,M)(\mathcal{F},L)\to (\mathcal{E},M) are pairs (p,f)(p,f) with p:p:\mathcal{F}\to\mathcal{E} a geometric morphism and f:p *MLf:p^*M\to L a 𝕋\mathbb{T}-model homomorphism, and

  • 2-cells (p,f)(q,g)(p,f)\to(q,g) are natural transformations η:pq\eta:p\to q such that f=gη Mf=g\circ\eta_M.

𝕋𝔗𝔬𝔭 N\mathbb{T}-\mathfrak{Top}_N is the full sub-2-category on pairs (,M)(\mathcal{E},M) such that \mathcal{E} has a natural numbers object.


Let 𝕋\mathbb{T} be a (geometric) quotient theory of 𝕊\mathbb{S}. A class 𝔸\mathbb{A} of 𝕋\mathbb{T}-model morphisms is called admissible if

  • 𝔸\mathbb{A} is closed under inverse image functors: p *f𝔸p^*f\in \mathbb{A} for f𝔸f\in \mathbb{A}.

  • 𝔸\mathbb{A} contains all identity morphisms and given g𝔸g\in \mathbb{A} and composable ff : f𝔸f\in \mathbb{A} iff gf𝔸gf\in \mathbb{A}.

  • Given an 𝕊\mathbb{S}-model morphism f:MLf:M\to L with LL a 𝕋\mathbb{T}-model, there exists a factorization MqM ff^LM\overset{q}{\to}M_f\overset{\hat{f}}{\to}L with M fM_f a 𝕋\mathbb{T}-model and f^𝔸\hat{f}\in \mathbb{A}, such that any other such factorization MrPpLM\overset{r}{\to}P\overset{p}{\to}L factors with gh=f^gh=\hat{f} and hq=rhq=r for a unique h:M fPh:M_f\to P. Moreover, this factorization is preserved by inverse image functors.

The sub-category 𝔸𝔗𝔬𝔭\mathbb{A}-\mathfrak{Top} of 𝕋𝔗𝔬𝔭\mathbb{T}-\mathfrak{Top} for such an admissible class has 1-cells (p,f)(p,f) with f𝔸f\in \mathbb{A}.


Let 𝕊\mathbb{S} and 𝕋\mathbb{T} be finitely presented geometric theories such that 𝕋\mathbb{T} is a quotient theory of 𝕊\mathbb{S}, and let 𝔸\mathbb{A} be an admissible class of morphisms of 𝕋\mathbb{T}-models. Then the inclusion functor 𝔸𝔗𝔬𝔭 N𝕊𝔗𝔬𝔭 N\mathbb{A}-\mathfrak{Top}_N\to \mathbb{S}-\mathfrak{Top}_N has a right adjoint Spec:𝕊𝔗𝔬𝔭 N𝔸𝔗𝔬𝔭 NSpec:\mathbb{S}-\mathfrak{Top}_N\to \mathbb{A}-\mathfrak{Top}_N.


The classical example is given by the geometric theories of commutative rings and local rings with the factorization given by the class of local morphisms and appropriate rings of fractions as the local factors. The right adjoint maps a commutative ring AA basically to the pair consisting of the sheaf topos on the Zariski spectrum of AA and the structure sheaf of AA (cf. Johnstone 1977b).


In the context of his work with C. Lair on ‘locally free diagrams’ R. Guitart interprets the ‘almost-freeness’ of the spec construction as an ‘almost-algebraicity’ of topology (See Guitart 2008 and the references therein).

spectrum of a commutative ring

locally algebra-ed topos

classifying topos

geometric theory

ringed topos

Structured Spaces

formally etale morphisms


The original article is reprinted as

  • Julian Cole, The bicategory of topoi and spectra, Reprints in Theory and Applications of Categories, No. 25 (2016) pp. 1-16 (TAC)

Besides this, Johnstone’s article (1977a), the book of Johnstone (1977b), on which the above exposition is based, is a good source for this material. Bunge (1981), Bunge and Reyes (1981) apply the results in a model-theoretic context. See also the short remark in Caramello (2014, p.55).

M. Coste‘s (1979) take on admissible maps flows in Coste-Michon (1981) via the 4-yoga (cf. above) into the more recent approach using open and étale maps pioneered by Joyal. Dubuc (2000) compares his axiomatic étale morphisms to Cole’s class of admissible morphisms.

For a glimpse of A. Joyal‘s original approach to the spectrum using distributive lattices see Joyal (1975), Español (1983,1986) or Coquand-Lombardi-Schuster (2007). For higher categorical variations on the theme spectrum (of a ring) see Lurie (2009).

  • M. Bunge, Sheaves and Prime Model Extensions , J. Algebra 68 (1981) pp.79-96.

  • M. Bunge, G. E. Reyes , Boolean spectra and model completions , Fund.Math. 113 (1981) pp.165-173. (pdf)

  • O. Caramello, Topos-theoretic background , ms. 2014. (pdf)

  • J. C. Cole, The Bicategory of Topoi, and Spectra , ms. (pdf)

  • Thierry Coquand, Henri Lombardi, Peter Schuster, The projective spectrum as a distributive lattice , Cah. Top. Géom. Diff. Cat. XLIIX no.3 (2007) pp.220-228. (numdam)

  • M. Coste, Localisation, spectra and sheaf representation , pp.212-238 in Fourman, Mulvey & Scott (eds.), Applications of Sheaves , Springer LNM 753 (1979).

  • M. Coste, G. Michon, Petits et Gros Topos en Géométrie Algébrique , Cah. Top. Géom. Diff. Cat. XXII no.1 (1981) pp.25-30. (pdf)

  • E. J. Dubuc, Axiomatic etal maps and a theory of spectrum , JPAA 149 (2000) pp.15-45.

  • L. Español, Le spectre d’un anneau dans l’algèbre constructive et applications à la dimension , Cah. Top. Géom. Diff. Cat. XXIV no.2 (1983) pp.133-144. (numdam)

  • L. Español, El Espectro y la Dimension des Anillos en el Algebra sobre un Topos , Revista Colombiana de Matemáticas XX (1986) pp.105-128. (gdz)

  • R. Guitart, Toute Théorie est Algébrique et Topologique , Cah.Top.Géom.Diff.Cat. XLIX no. 2 (2008) pp.83-128. (pdf)

  • M. Hakim, Topos Annelés et Schémas Relatifs , Springer Heidelberg 1972.

  • P. T. Johnstone, Rings, fields and spectra , J. Algebra 49 (1977) pp.238-260.

  • P. T. Johnstone, Topos Theory , Academic Press New York 1977 (Dover reprint 2014). pp.205-207

  • A. Joyal, Les Théorèmes de Chevalley-Tarski et Remarques sur l’Algèbre Constructive , Cah. Top. Géom. Diff. Cat. XVI (1975) pp.256-258. (Proceedings Colloque Amiens 1975, 6.81 MB)

  • J. Lurie, Derived Algebraic Geometry V: Structured Spaces , ms. 2009. (pdf)

  • R. O. Robson, Model theory and spectra , JPAA 63 (1990) pp.301-327.

  1. Coste&Michon (1981), p.27.

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