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\section*{big and little toposes}

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

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

[[!include topos theory - contents]]

\hypertarget{big_and_little_toposes}{}\section*{{Big and little toposes}}\label{big_and_little_toposes}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{relationships}{Relationships}\dotfill \pageref*{relationships} \linebreak
\noindent\hyperlink{the_big_and_little_topos_of_an_object}{The big and little topos of an object}\dotfill \pageref*{the_big_and_little_topos_of_an_object} \linebreak
\noindent\hyperlink{axiomatizations}{Axiomatizations}\dotfill \pageref*{axiomatizations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

There are two different (related) relationships between [[Grothendieck topoi]] and a notion of \emph{generalized [[space]]}. (Recall that a Grothendieck topos $T$ is a [[category of sheaves]] $T = Sh(S)$ on some [[site]] $S$.)

On the one hand, we can regard the topos \emph{itself} as a generalized space. This tends to be a useful point of view when the site $S$ is the [[category of open subsets]] $Op(X)$ of a [[topological space]] $X$ (or some [[manifold]] or the like), or some other site which we regard as containing data from only ``one space.'' In this case, we refer to $T$ as a \textbf{little topos}, or (if we fail to translate the original French) a \textbf{petit topos}.

On the other hand, we can view a topos $T$ as a well-behaved category whose \emph{objects} are generalized spaces. This tends to be a useful point of view when the site $S$ is a category of \emph{all test [[space]]s} in some sense, such as [[Top]], [[Diff]], or [[CartSp]]. In this case, we refer to $T$ as a \textbf{big topos}, or (in French) a \textbf{gros topos}.

These distinctions carry over in a straightforward way to higher topoi such as [[(∞,1)-topoi]].

\hypertarget{relationships}{}\subsection*{{Relationships}}\label{relationships}

Objects in a big topos $Sh(S)$ may be thought of as [[space]]s \emph{modeled on $S$}, in the sense described at [[motivation for sheaves, cohomology and higher stacks]] and at [[space]].

On the other hand, the objects of a petit topos, such as $Sh(X)$, can also be regarded as a kind of generalized spaces, but generalized spaces \emph{over $X$} on which the rigid structure of morphisms in $Op(X)$ (only inclusions of subsets, no more general maps) induces a correspondingly rigid structure so that they are not all that general. In fact, $Sh(Op(X))$ is equivalent to the category of [[etale space]]s over $X$---i.e. spaces ``modeled on $X$'' in a certain sense. More generally, for any topos $E$, the objects of $E$ can be identified with [[local homeomorphisms of toposes]] into $E$.

From the ``little topos'' perspective, it can be helpful to think of a ``big topos'' as a ``fat point,'' which is not ``spread out'' very much spatially itself, but contains within that point lots of different types of ``local data,'' so that even spaces which are ``rigidly'' modeled on that point can have a lot of interesting cohesion and local structure. (One should not be misled by this into thinking that a big topos has \emph{only} one [[point of a topos|point]], although it is usually a [[local topos]] and hence has an [[initial object|initial]] point.)

\hypertarget{the_big_and_little_topos_of_an_object}{}\subsection*{{The big and little topos of an object}}\label{the_big_and_little_topos_of_an_object}

If $X$ is a topological space, then the canonical little topos associated to $X$ is the sheaf topos $Sh(X)$. On the other hand, if $S$ is a site of probes enabling us to regard $X$ as an object of a big topos $H = Sh(S)$, then we can also consider the topos $H/X$ as a representative of $X$. These two toposes are often called the \textbf{little topos of $X$} (or \textbf{petit topos of $X$}) and the \textbf{big topos of $X$} (or \textbf{gros topos of $X$}) respectively.

There might be some debate about whether $H/X$ is, itself, ``a little topos'' or ``a big topos.'' While it certainly contains information about the space $X$ specifically, its objects are not ``spaces locally modeled on $X$'' but rather spaces locally modeled on the big site $S$ which happen to have a map to $X$. The standard phrase ``the big topos of $X$'' is the most descriptive.

Note that if $X$ is actually an \emph{object} of the site $S$, then $H/X$ can be identified with the topos of sheaves on the [[slice category|slice]] site $S/X$ (and otherwise, it can be identified with the topos of sheaves on the [[category of elements]] of $X\in Sh(S)$). This site $S/X$ is often referred to as the [[big site]] of $X$, as compared to the [[little site]], which is $Op(X)$ (or appropriate replacement). The topos $Sh(S/X)$ can thus be viewed as spaces modelled on $S$, but parameterised by the representable sheaf $X$.

Note that when $S=Top$ with its local-homeomorphism topology, there is a canonical functor $Op(X) \to S/X$ which preserves finite limits and both [[cover-preserving functor|preserves]] and [[cover-reflecting functor|reflects]] covering families. Therefore, it induces both a geometric morphism $H/X \to Sh(X)$ and one $Sh(X) \to H/X$, of which the latter is the left adjoint of the former in [[Topos]]. In other words, the geometric morphism $H/X \to Sh(X)$ is [[local geometric morphism|local]], and in particular a [[homotopy equivalence of toposes]]. This fact relating the big and little toposes of $X$ also holds in other cases.

\hypertarget{axiomatizations}{}\subsection*{{Axiomatizations}}\label{axiomatizations}

\begin{itemize}%
\item If a site $S$ is given by a [[Grothendieck pretopology]], then one can define an associated notion of a [[little site]] associated to any object of $S$, and hence both a little topos and a big topos, which are related as above.


\item One proposed axiomatization of the notion of big topos is that of a [[cohesive topos]].


\item In his early papers in the 80s, [[Lawvere]] emphasized the existence of a contractible [[subobject classifier]], a concept which together with the [[adjoint quadruple]] goes under the name [[sufficiently cohesive topos]] in the later axiomatization (modulo some fineprint).



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

For $X$ a [[topological space]], the \emph{little topos} that it defines is the [[category of sheaves]] $Sh(X) := Sh(Op(X))$ on the [[category of open subsets]] of $X$. A general object in this topos can be regarded as an [[etale space]] over $X$. The space $X$ itself is incarnated as the [[terminal object]] $X = * \in Sh(X)$.

On the other hand, a \emph{big topos} in which $X$ is incarnated is a [[category of sheaves]] on a [[site]] of test spaces with which $X$ may be probed. For instance for $C =$ [[Top]], or [[Diff]] or [[CartSp]] with their standard [[coverage]]s, $Sh(C)$ is such a big topos. See for instance, [[topological topos]] and the [[quasi-topos]] of [[quasitopological space]]s.

In good cases, the intrinsic properties of $X$ do not depend on whether one regards it as a little topos or as an object of a gros topos. For instance at [[cohomology]] in the section  it is discussed how the [[nonabelian cohomology]] of a [[paracompact space|paracompact]] [[manifold]] $X$ with constant coefficients gives the same answer in each case.

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

\begin{itemize}%
\item [[functorial geometry]], [[space and quantity]]


\item [[topological site]], [[continuous truth]]


\item [[cohesive topos]]

\begin{itemize}%
\item [[sufficiently cohesive topos]]


\item [[infinitesimal cohesive (infinity,1)-topos]]


\item [[quality type]]



\end{itemize}

\item [[étendue]]


\item [[locally decidable topos]]



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

The notion of a \emph{gros topos} of a \emph{topological space} is due to [[Jean Giraud]]. Some early results from the Grothendieck school appear in

\begin{itemize}%
\item [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], \emph{Th\'e{}orie des Topos et Cohomologie Etale des Sch\'e{}mas ([[SGA4]])} , Springer LNM \textbf{269} (1972). (expos\'e{} IV, 2.5 pp.316-318, 4.10 pp.358-365)

\end{itemize}
In this context see also

\begin{itemize}%
\item [[Saunders Mac Lane|S. Mac Lane]], [[Ieke Moerdijk|I. Moerdijk]], \emph{Sheaves in Geometry and Logic} , Springer Heidelberg 1994. (pp.113, 325, 416)

\end{itemize}
In the context of a discussion of the [[big Zariski topos]] Lawvere calls the gros-petit distinction `\emph{a surprising twist of logic that is not yet fully clarified}' on p.110 of his contribution to the \emph{Eilenberg-Festschrift}:

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Variable quantities and variable structures in topoi} , pp.101-131 in Heller, Tierney (eds.), \emph{Algebra, Topology and Category Theory} , Academic Press New York 1976.

\end{itemize}
The suggestion that a \emph{general notion} of gros topos is needed goes back to some remarks in \emph{[[Pursuing Stacks]]}. A precise axiom system capturing the notion is first proposed in

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Categories of spaces may not be generalized spaces, as exemplified by directed graphs}, preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1--7.(\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.144.6357&rep=rep1&type=pdf}{pdf})

\end{itemize}
The axioms 0 and 1 for \emph{toposes of generalized spaces} given there later became called the axioms for a [[cohesive topos]], together with axiom 2 they make out a [[sufficiently cohesive topos]].

Further discussion of this axiomatics for [[gros topos]]es is in

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Categories of space and quantity} in: J. Echeverria et al (eds.), \emph{The Space of mathematics}, de Gruyter, Berlin, New York (1992)

\end{itemize}
where a proposal for a general axiomatization of [[homotopy]]/[[homology]]-like ``extensive quantities'' and [[cohomology]]-like ``intensive quantities'') as covariant and contravariant functors out of a distributive category are considered.

The following two papers contain Lawvere's early view of a trichotomy between big toposes vs. étendue and locally decidable toposes as paradigmatic ``generalized spaces'' with ``infinitesimally cohesive'' in between, with the latter subsumed into the fine structure of cohesion in more recent versions

\begin{itemize}%
\item [[F. W. Lawvere]], \emph{Qualitative Distinctions between some Toposes of Generalized Graphs} , Cont. Math. \textbf{92} (1989) pp.261-299.


\item [[F. W. Lawvere]], \emph{[[Some Thoughts on the Future of Category Theory]]} , pp.1-13 in Springer LNM \textbf{1488} (1991).



\end{itemize}
The left and right adjoint to the global section functor as a means to identify discrete and codiscrete spaces respectively is also mentioned in

\begin{itemize}%
\item [[Bill Lawvere]] \emph{Taking categories seriously}, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1--24. (\href{http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf}{pdf})

\end{itemize}
on \href{http://www.tac.mta.ca/tac/reprints/articles/8/tr8.pdf#page=14}{page 14}.

Under the term \emph{categories of cohesion} these axioms are discussed in

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Axiomatic cohesion} Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41--49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf})

\end{itemize}
Another definition of gros vs petit toposes and remarks on applications in [[Galois theory]] is in

\begin{itemize}%
\item Nick Duncan, \emph{Gros and petit toposes} (\href{http://www.cheng.staff.shef.ac.uk/pssl88/pssl88-duncan.pdf}{pdf})

\end{itemize}
and yet another one is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Spectral Algebraic Geometry]]}, chapter 20 ``Fractured $\infty$-Topoi'' (\href{http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf}{pdf})

\end{itemize}
There is also something relevant in this article:

\begin{itemize}%
\item [[Mathieu Anel]], \emph{Grothendieck topologies from unique factorization systems} (\href{http://arxiv.org/abs/0902.1130}{arXiv:0902.1130})


\item [[Mamuka Jibladze]], \emph{Homotopy types for ``gros'' toposes}, thesis, \href{http://www.rmi.ge/~jib/pubs/thesis.pdf}{pdf}


\item [[Peter Johnstone]], \emph{Calibrated Toposes} , Bull. Belgian Math. Soc. - Simon Stevin \textbf{19} no.5 (2012) pp.889-907. (\href{http://projecteuclid.org/euclid.bbms/1354031555}{projecteuclid})



\end{itemize}
A discussion and comparison of big vs little approaches to $(\infty,1)$-topos theory began at these blog entries:

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/category/2010/10/cohesive_toposes.html}{Cohesive (∞,1)-toposes} and \href{http://golem.ph.utexas.edu/category/2010/10/petit_1toposes.html}{Petit (∞,1)-toposes}.

\end{itemize}
[[!redirects big topos]] [[!redirects big toposes]] [[!redirects big topoi]] [[!redirects gros topos]] [[!redirects gros toposes]] [[!redirects gros topoi]]

[[!redirects little topos]] [[!redirects little toposes]] [[!redirects little topoi]] [[!redirects petit topos]] [[!redirects petit toposes]] [[!redirects petit topoi]]

[[!redirects big or little topos]] [[!redirects big or little toposes]] [[!redirects big or little topoi]] [[!redirects big and little topos]] [[!redirects big and little toposes]] [[!redirects big and little topoi]] [[!redirects little or big topos]] [[!redirects little or big toposes]] [[!redirects little or big topoi]] [[!redirects little and big topos]] [[!redirects little and big toposes]] [[!redirects little and big topoi]] [[!redirects gros or petit topos]] [[!redirects gros or petit toposes]] [[!redirects gros or petit topoi]] [[!redirects gros and petit topos]] [[!redirects gros and petit toposes]] [[!redirects gros and petit topoi]] [[!redirects petit or gros topos]] [[!redirects petit or gros toposes]] [[!redirects petit or gros topoi]] [[!redirects petit and gros topos]] [[!redirects petit and gros toposes]] [[!redirects petit and gros topoi]]

[[!redirects petit (∞,1)-topos]] [[!redirects petit (infinity,1)-topos]] [[!redirects petit (∞,1)-toposes]] [[!redirects petit (infinity,1)-toposes]]

[[!redirects gros (∞,1)-topos]] [[!redirects gros (infinity,1)-topos]] [[!redirects gros (∞,1)-toposes]] [[!redirects gros (infinity,1)-toposes]]



\end{document}