The bornological topos is a Grothendieck topos based on the concept of a bounded sequence and was proposed by William Lawvere in unpublished lectures in 1983 as a convenient category to do functional analysis in.
Let $\mathcal{C}$ be the category of countable sets. The bornological topos $\mathcal{B}$ is the category of finite product preserving presheaves on $\mathcal{C}$.
Equivalently, $\mathcal{B}$ is the topos of sheaves on $\mathcal{C}$ for the (finite) disjoint covering topology with coverings the finite families $(X_i\to Y)_{i\in I}$ , such that $\sum_{i\in I} X_i\to Y$ is an isomorphism.
In the work of Español et al., $\mathcal{B}$ is accessed as a subtopos of the topos of actions of the monoid of endomorphisms of $N$.
$\mathcal{B}$ contains the category of (Kolmogorov) bornological spaces as a full reflective subcategory (cf. Español-Lambán 2002).
The object $R_D$ of Dedekind real numbers of $\mathcal{B}$ is the space $l^\infty$ of bounded sequences of real numbers (cf. Español-Mínguez 2001).
“ The category of modules over the Dedekind real number object includes all the inductive limits of Banach spaces as a full subcategory, but at the same time is itself a Grothendieck AB5 Abelian category with the accompanying exactness properties. ” (Lawvere 2008, p.15). Thus “ functional analysis becomes linear algebra ” in the bornological topos (Lawvere 1994, p.10).
The bornological topos is the Gaeta topos on $\mathcal{C}$ and as such fits Lawvere’s paradigm of doing abstract “algebraic geometry” (cf. Lawvere 1986, p.17). In particular, the geometry and cohesiveness of the objects $X$ in $\mathcal{B}$ arises covariantly from the basic figure shape of a ‘bounded’ sequence $A$ via maps $A\to X$.
That bornology provides in the context of functional analysis “a more basic notion of cohesiveness” than the usual topological neighborhood concept and contravariant function algebra concept based on it is argued in Lawvere (1997, pp.3f).
Although the bornological topos can be regarded as a cohesive category of “spaces” in a broad sense, it doesn’t satisfy Lawvere’s axiomatic cohesion since it lacks the required left adjoint components functor $\Pi:\mathcal{B}\to Set$ (cf. Lawvere 2008).
L. Español, Extended real number object in the bornological topos , talk CT07 Carvoeiro 2007. (pdf-slides)
L. Español, L. Lambán, A tensor-hom adjunction in a topos related to vector topologies and bornologies , Journal of Pure and Applied Algebra 154 (2000) pp.143-158. (doi:10.1016/S0022-4049(99)00188-7)
L. Español, L. Lambán, On bornologies, locales and toposes of M-sets , Journal of Pure and Applied Algebra 176 (2002) pp.113–125. (doi:10.1016/S0022-4049(02)00047-6)
L. Español, M. C. Mínguez, Cortaduras Para $l^\infty$ , pp.375-390 in Español, Varona (eds.), Margarita Mathematica en Memoria de José Javier (Chicho) Guadalupe Hernández , Universidad de La Rioja 2001. (pdf)
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, Qualitative Distinctions between some Toposes of Generalized Graphs , Contemporary Mathematics 92 (1989) pp.261-299. (Google Books link; doi:10.1090/conm/092/1003203)
F. William Lawvere, Cohesive Toposes and Cantor’s ‘lauter Einsen’ , Phil. Math. 2 no.3 (1994) pp.5-15.
F. William Lawvere, Volterra’s functionals and covariant cohesion of space , in Proceedings of the May 1997 Meeting in Perugia , Perugia Studies in Mathematics 1997.
F. William Lawvere, Cohesive Toposes: Combinatorial and Infinitesimal Cases, Como Ms. 2008. (pdf)
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