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\begin{document}

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\section*{bornological topos}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{remark_bornology_and_cohesiveness}{Remark: bornology and cohesiveness}\dotfill \pageref*{remark_bornology_and_cohesiveness} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{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is a [[Grothendieck topos]] based on the concept of a bounded sequence and was proposed by [[William Lawvere]] in unpublished lectures in 1983 as a convenient category to do [[functional analysis]] in.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\mathcal{C}$ be the category of [[countable set|countable sets]]. The \emph{bornological topos} $\mathcal{B}$ is the category of finite product preserving presheaves on $\mathcal{C}$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item Equivalently, $\mathcal{B}$ is the topos of sheaves on $\mathcal{C}$ for the \emph{(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.


\item In the work of Espa\~n{}ol et al., $\mathcal{B}$ is accessed as a subtopos of the topos of actions of the monoid of endomorphisms of $N$.


\item $\mathcal{B}$ contains the category of (Kolmogorov) [[bornological set|bornological spaces]] as a full reflective subcategory (cf. Espa\~n{}ol-Lamb\'a{}n \hyperlink{EspanolLamban02}{2002}).


\item The object $R_D$ of [[Dedekind real numbers]] of $\mathcal{B}$ is the space $l^\infty$ of bounded sequences of real numbers (cf. Espa\~n{}ol-M\'i{}nguez \hyperlink{Espanol-Minguez01}{2001}).


\item `` \emph{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 \hyperlink{Lawvere08}{2008}, p.15). Thus `` \textbf{functional analysis becomes linear algebra} '' in the bornological topos (Lawvere \hyperlink{Lawvere94}{1994}, p.10).



\end{itemize}
\hypertarget{remark_bornology_and_cohesiveness}{}\subsection*{{Remark: bornology and cohesiveness}}\label{remark_bornology_and_cohesiveness}

The bornological topos is the [[Gaeta topos]] on $\mathcal{C}$ and as such fits Lawvere's paradigm of doing abstract ``algebraic geometry'' (cf. Lawvere \hyperlink{Lawvere86}{1986}, p.17). In particular, the geometry and cohesiveness of the objects $X$ in $\mathcal{B}$ arises \emph{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 ``\emph{a more basic notion of cohesiveness}'' than the usual topological neighborhood concept and contravariant function algebra concept based on it is argued in Lawvere (\hyperlink{Lawvere97}{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 \emph{axiomatic cohesion} since it lacks the required left adjoint components functor $\Pi:\mathcal{B}\to Set$ (cf. Lawvere \hyperlink{Lawvere08}{2008}).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[bornology]]


\item [[Johnstone's topological topos|topological topos]]


\item [[topos of recursive sets|recursive topos]]


\item [[cohesion]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item L. Espa\~n{}ol, \emph{Extended real number object in the bornological topos} , talk CT07 Carvoeiro 2007. (\href{http://www.mat.uc.pt/~categ/ct2007/slides/espanol.pdf}{pdf-slides})


\item L. Espa\~n{}ol, L. Lamb\'a{}n, \emph{A tensor-hom adjunction in a topos related to vector topologies and bornologies} , Journal of Pure and Applied Algebra \textbf{154} (2000) pp.143-158. (doi:\href{http://dx.doi.org/10.1016/S0022-4049(99%2900188-7}{10.1016/S0022-4049(99)00188-7})


\item L. Espa\~n{}ol, L. Lamb\'a{}n, \emph{On bornologies, locales and toposes of M-sets} , Journal of Pure and Applied Algebra \textbf{176} (2002) pp.113--125. (doi:\href{http://dx.doi.org/10.1016/S0022-4049(02%2900047-6}{10.1016/S0022-4049(02)00047-6})


\item L. Espa\~n{}ol, M. C. M\'i{}nguez, \emph{Cortaduras Para $l^\infty$} , pp.375-390 in Espa\~n{}ol, Varona (eds.), \emph{Margarita Mathematica en Memoria de Jos\'e{} Javier (Chicho) Guadalupe Hern\'a{}ndez} , Universidad de La Rioja 2001. (\href{http://www.emis.de/proceedings/Chicho2001/Espanol-Minguez.pdf}{pdf})


\item [[F. William Lawvere]], \emph{Taking Categories Seriously}, Revista Colombiana de Matem\'a{}ticas \textbf{XX} (1986) pp.147-178. Reprinted as TAC Reprint no.8 (2005) pp.1-24. (\href{ftp://ftp.tac.mta.ca/tac/html/tac/reprints/articles/8/tr8.pdf}{pdf})


\item [[F. William Lawvere]], \emph{Qualitative Distinctions between some Toposes of Generalized Graphs} , Contemporary Mathematics \textbf{92} (1989) pp.261-299. (\href{https://books.google.com.au/books?id=VxAcCAAAQBAJ&pg=PA261}{Google Books link}; doi:\href{http://dx.doi.org/10.1090/conm/092/1003203}{10.1090/conm/092/1003203})


\item [[F. William Lawvere]], \emph{Cohesive Toposes and Cantor's `lauter Einsen'} , Phil. Math. \textbf{2} no.3 (1994) pp.5-15.


\item [[F. William Lawvere]], \emph{Volterra's functionals and covariant cohesion of space} , in \emph{Proceedings of the May 1997 Meeting in Perugia} , Perugia Studies in Mathematics 1997.


\item [[F. William Lawvere]], \emph{Cohesive Toposes: Combinatorial and Infinitesimal Cases}, Como Ms. 2008. (\href{http://comocategoryarchive.com/Archive/temporary_new_material/FWLawvere-Cohesive-Toposes-Como-January-2008.pdf}{pdf})



\end{itemize}
[[!redirects Bornological topos]]



\end{document}