nLab Gaeta topos

Idea

The Gaeta topos construction assigns a Grothendieck topos to a small extensive category.

The construction arises in attempts (Lawvere 1976, 1991) to build a gros topos of spaces with an inherent space-quantity ‘duality’. The terminology is apparently a tribute to the Spanish algebraic geometer Federico Gaeta? who in his lecture notes of A. Grothendieck's Buffalo 1973 colloquium (Gaeta 1974) has pointed out the utility of the extensivity property for functorial algebraic geometry.¹

Definition

Let $\mathcal{C}$ be a small finitary extensive category. The Gaeta topology² on $\mathcal{C}$ is the Grothendieck topology $J_G$ generated by all finite families $(X_i\to X)$ such that $\sum_i X_i\to X$ is an isomorphism. The Gaeta topos $G(\mathcal{C})$ is the topos of sheaves for the site $(\mathcal{C}, J_G)$.

Examples

• The bornological topos is the Gaeta topos on the category of countable sets.

Properties

The Gaeta topology $J_G$ is naturally induced by the extensivity of $\mathcal{C}$:

Proposition

The Gaeta topology $J_G$ as defined above is indeed a Grothendieck topology on a small extensive category $\mathcal{C}$.

Proof

…still to do!

The next proposition shows that $G(\mathcal{C})$ is not only a topos but a multisorted variety as well as it coincides with the category of finite product preserving presheaves on $\mathcal{C}$:

Proposition

Let $\mathcal{C}^{op}$ be a small Cauchy complete category with finite products. The category $Prod(\mathcal{C}^{op},Set)$ of finite product preserving functors $\mathcal{C}^{op}\to Set$ is a topos iff $\mathcal{C}$ is extensive. Moreover, in this case $Prod(\mathcal{C}^{op},Set)$ coincides with $G(\mathcal{C})$.

Cf. Carboni-Pedicchio-Rosicky (2001, p.72) and Lawvere (2008, p.498).

From an algebraic perspective, $Prod(\mathcal{C}^{op},Set)$ is the category $\mathcal{C}^{op}-Mod$ of algebras of the multisorted algebraic theory $\mathcal{C}^{op}$ .

Note, since they are naturally equivalent to $Hom_\mathcal{C}(-,X)$ which preserve all limits (cf. hom-functor), representable presheaves preserve finite products, in particular, and hence are sheaves for $J_G$; in other words, the Gaeta topology is subcanonical.

Buffalo style algebraic geometry

…

Related entries

• space and quantity

• functorial geometry

• bornological topos

• extensive category

• superextensive site

• gros topos

• Nullstellensatz

• Zariski topos

Link

• F. William Lawvere, Grothendieck’s 1973 Buffalo Colloquium, posting to the mailing list [e-mail], march 2003. (link)

References

• Aurelio Carboni, Maria Cristina Pedicchio?, Jiri Rosicky, Syntactic characterization of various classes of locally presentable categories , JPAA 161 (2001) pp.65-90.(zbMath entry)

• Federico Gaeta?, Introduction to Functorial Algebraic Geometry - After a Summer Course by A. Grothendieck: I Affine Algebraic Geometry , Suny Buffalo 1974. (pdf scan)

• F. William Lawvere, Variable quantities and variable structures in topoi , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976. (pp.109-111)

• F. William Lawvere, Taking Categories Seriously, Revista Colombiana de Matemáticas XX (1986) pp.147-178. Reprinted as TAC Reprint no.8 (2005) pp.1-24. (pdf)

• F. William Lawvere, Some Thoughts on the Future of Category Theory , pp.1-13 in LNM 1488 Springer Heidelberg (1991). (pp.3-6)

• F. William Lawvere, Comments on the development of topos theory , pp.715-734 in Jean-Pierre Pier (ed.), Development of Mathmatics 1950-2000 , Birkhäuser Basel 2000. Reprinted with author comment as TAC Reprint no.24 (2014). (abstract; p.727)

• F. William Lawvere, Core Varieties, Extensivity, and Rig Geometry , TAC 20 no.14 (2008) pp. 497–503. (pdf)

• F. Marmolejo, M. Menni, Level $\epsilon$ , arXiv:1909.12757 (2019). (abstract)

• Matías Menni, Sufficient Cohesion over Atomic Toposes , Cah. Top. Géom. Diff. Cat. LV no.2 (2014) pp.113-149. (preprint)

• Matías Menni, Decidable objects and molecular toposes (2022) [arXiv:arXiv:2212.03647]

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1. “Gaeta’s notes of Grothendieck’s lecture series at Buffalo point out that $A$ (i.e. the opposite category of finitely presented algebras over a field $k$ -nLab) is more closely suited than most categories to serve as a site for a geometric category, because it is what is now called ‘extensive’ “ , (Lawvere 2003). ↩

2. $J_G$ is also called the finite disjoint covering topology. The nLab elsewhere uses the term extensive topology (cf. extensive category). $(\mathcal{C},J_G)$ is in fact an example of a superextensive site. ↩