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\section*{ETCS}

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

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

[[!include topos theory - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - 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{a_constructive_view}{A constructive view}\dotfill \pageref*{a_constructive_view} \linebreak
\noindent\hyperlink{a_contemporary_perspective}{A contemporary perspective}\dotfill \pageref*{a_contemporary_perspective} \linebreak
\noindent\hyperlink{ExpositionByTrimble}{Todd Trimble's exposition of ETCS}\dotfill \pageref*{ExpositionByTrimble} \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{Elementary Theory of the Category of Sets} , or \emph{ETCS} for short, is an axiomatic formulation of [[set theory]] in a [[category theory|category-theoretic]] spirit. As such, it is the prototypical [[structural set theory]]. Proposed shortly after [[ETCC]] in (\hyperlink{Lawvere64}{Lawvere 64}) it is also the paradigm for a [[foundations|categorical foundation]] of mathematics.\footnote{For a comparative discussion of its virtues as foundation see [[foundations of mathematics]] , the \hyperlink{ExpositionByTrimble}{texts by Todd Trimble} or the informative paper by \hyperlink{McLarty04}{McLarty (2004)}.} 

The theory intends to capture in an invariant way the notion of a (constant) \emph{`abstract set'} whose elements lack internal structure and whose only external property is cardinality with further external relations arising from \emph{mappings}. The membership relation is \emph{local} and \emph{relative} i.e. membership is meaningful only between an element of a set and a subset of the very same set. (See \hyperlink{Lawvere76}{Lawvere (1976, p.119)} for a detailed description of the notion `abstract set'.\footnote{It has been pointed out by John Myhill that Cantor's concept of `cardinal' as a set of abstract units should be viewed as a structural set theory and a precursor to Lawvere's concept of an `abstract set'. This view is endorsed and expanded in \hyperlink{Lawvere94}{Lawvere 1994}.}  \footnote{[[Richard Dedekind| R. Dedekind's]] views are also anticipating `abstract sets' e.g. Bernstein reports in Dedekind's works vol.3 (1932, p.449) that Dedekind gave as his intuition of a set: ``a closed bag, containing determinate things that one can not see and of which one knows nothing beyond their existence and determinateness''.}  \footnote{The first axiomatic set theory without primitive membership relation $\in$ was presumably proposed by A. Schoenflies in 1920: he modeled elements of sets as indecomposable subsets. See A. Schoenflies, \emph{Zur Axiomatik der Mengenlehre} , Math. Ann. \textbf{83} (1921) pp.173-200; and \emph{Bemerkung zur Axiomatik der Gr\"o{}ssen und Mengen} , Math. Ann. \textbf{85} (1922) pp.60-64.}  \footnote{The first axiomatic set theory based on the notion of function was [[John von Neumann|von Neumann]]`s 1925 version of what later became the set based [[NBG]] theory of classes.} )

More in detail, ETCS is a [[first-order theory]] axiomatizing [[elementary toposes]] and specifically those which are [[well-pointed topos|well-pointed]], have a [[natural numbers object]] and satisfy the [[axiom of choice]]. The theory omits the [[axiom of replacement]], however.

The idea is, first of all, that much of traditional mathematics naturally takes place ``[[internal logic|inside]]'' such a topos of \emph{constant} sets, and second that this perspective generalizes beyond ETCS proper to toposes of \emph{variable} and \emph{cohesive} sets by varying the axioms: for instance omitting the [[well-pointed topos|well-pointedness]] and the [[axiom of choice]] but adding the [[Kock-Lawvere axiom]] gives a [[smooth topos]] inside which [[synthetic differential geometry]] takes place.

That is, ETCS locates the category of sets by the well-pointedness axiom as the discrete zero point on a `continuous' range of toposes eligible for foundations. In particular, whereas ZF mainly provides `substance' for mathematics, ETCS lives as a special type of form within the continuum of mathematical form itself.

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

The axioms of ETCS can be summed up in one sentence as:

\begin{defn}
\label{}\hypertarget{}{}
The [[Set|category of sets]] is [[generalized the|the]] [[topos]] which

\begin{enumerate}%
\item is a [[well-pointed topos]]


\item has a [[natural numbers object]]


\item and satisfies the [[axiom of choice]].



\end{enumerate}
\end{defn}
For more details see

\begin{itemize}%
\item [[fully formal ETCS]].

\end{itemize}
\hypertarget{a_constructive_view}{}\subsection*{{A constructive view}}\label{a_constructive_view}

[[Erik Palmgren]] (\hyperlink{Palmgren}{Palmgren 2012}) has a [[constructive mathematics|constructive]] [[predicative mathematics|predicative]] variant of ETCS, which can be summarized as:

$Set$ is a [[well-pointed topos|well-pointed]] $\Pi$-[[Π-pretopos|pretopos]] with a [[NNO]] and [[enough projectives]] (i.e. [[COSHEP]] is satisfied). Here ``well-pointed'' must be taken in its constructive sense, as including that the [[terminal object]] is indecomposable and projective.

\hypertarget{a_contemporary_perspective}{}\subsection*{{A contemporary perspective}}\label{a_contemporary_perspective}

Modern mathematics with its emphasis on concepts from [[homotopy theory]] would more directly be founded in a similar spirit by an axiomatization not just of [[elementary toposes]] but of [[elementary (∞,1)-toposes]]. This is roughly what [[univalence|univalent]] [[homotopy type theory]] accomplishes -- for more on this see at \emph{\href{relation+between+type+theory+and+category+theory#HomotopyWithUnivalence}{relation between type theory and category theory -- Univalent HoTT and Elementary infinity-toposes}}.

Instead of increasing the [[higher category theory|higher categorical dimension]] [[(n,r)-category|(n,r)]] in the first argument, one may also, in this context of elementary foundations, consider raising the second argument. The case $(2,2)$ is the elementary theory of the 2-category of categories ([[ETCC]]).

\hypertarget{ExpositionByTrimble}{}\subsection*{{Todd Trimble's exposition of ETCS}}\label{ExpositionByTrimble}

[[Todd Trimble]] has a series of expository writings on ETCS which provide a very careful introduction and at the same time a wealth of useful details.

\begin{itemize}%
\item Todd Trimble, \emph{ZFC and ETCS: Elementary Theory of the Category of Sets} ([[Trimble on ETCS I|nLab entry]], \href{http://topologicalmusings.wordpress.com/2008/09/01/zfc-and-etcs-elementary-theory-of-the-category-of-sets/}{original blog entry})


\item Todd Trimble, \emph{ETCS: Internalizing the logic} ([[Trimble on ETCS II|nLab entry]], \href{http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic/}{original blog entry})


\item Todd Trimble, \emph{ETCS: Building joins and coproducts} ([[Trimble on ETCS III|nLab entry]], \href{http://topologicalmusings.wordpress.com/2008/12/15/etcs-building-joins-and-coproducts/}{original blog entry})



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

\begin{itemize}%
\item [[fully formal ETCS]]
\item [[structural set theory]]
\item [[ZFC|Zermelo Fraenkel set theory]]
\item [[Cohesive Toposes and Cantor's ``lauter Einsen'']]
\item [[axiom of replacement]]
\item [[ETCC]]
\item [[practical foundations of mathematics]]
\item [[foundations of mathematics]]

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

ETCS grew out of Lawvere's experiences of teaching undergraduate foundations of analysis at Reed college in 1963 and was originally published in

\begin{itemize}%
\item [[William Lawvere]], \emph{An elementary theory of the category of sets} , Proceedings of the National Academy of Science of the U.S.A \textbf{52} pp.1506-1511 (1964). (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC300477/pdf/pnas00186-0196.pdf}{pdf})

\end{itemize}
A more or less contemporary review is

\begin{itemize}%
\item C.C. Elgot, \emph{Review}, JSL \textbf{37} no.1 (1972) pp. 191-192.

\end{itemize}
A longer version of Lawvere's 1964 paper appears in

\begin{itemize}%
\item [[William Lawvere]], [[Colin McLarty]], \emph{An elementary theory of the category of sets (long version) with commentary} , Reprints in Theory and Applications of Categories \textbf{11} (2005) pp. 1-35. (\href{http://tac.mta.ca/tac/reprints/articles/11/tr11abs.html}{TAC})

\end{itemize}
An undergraduate set-theory textbook using it is

\begin{itemize}%
\item [[William Lawvere]], [[Robert Rosebrugh]], \emph{[[Sets for Mathematics]]} , CUP 2003. (\href{http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics}{web})

\end{itemize}
Lawvere explains in detail his views on constant and variable `abstract sets' on pp.118-128 of

\begin{itemize}%
\item [[William Lawvere]], \emph{Variable Quantities and Variable Structures in Topoi} , pp.101-131 in Heller, Tierney (eds.), \emph{Algebra, Topology and Category Theory: a Collection of Papers in Honor of Samuel Eilenberg} , Academic Press New York 1976.

\end{itemize}
See also ch. 2,3 of

\begin{itemize}%
\item [[William Lawvere]], \emph{Variable Sets Entendu and Variable Structure in Topoi} , lecture notes University of Chicago 1975.

\end{itemize}
On the anticipation of `abstract sets' in [[Georg Cantor|Cantor]]:

\begin{itemize}%
\item [[William Lawvere]], \emph{[[Cohesive Toposes and Cantor's ``lauter Einsen'']]}, Philosophia Mathematica \textbf{2} no.3 (1994) pp.5-15. ([[LawvereCohesiveToposes.pdf:file]])

\end{itemize}
A short overview article on ETCS:

\begin{itemize}%
\item [[Tom Leinster]], \emph{Rethinking set theory} \href{http://arxiv.org/abs/1212.6543}{arXiv}.

\end{itemize}
An insightful and non-partisan view of ETCS can be found in a section of:

\begin{itemize}%
\item [[Andreas Blass]], Yuri Gurevich, \emph{Why Sets ?} , Bull. Europ. Assoc. Theoret. Comp. Sci. \textbf{84} (2004) 139-156. (\href{http://www.math.lsa.umich.edu/~ablass/set.pdf}{draft})

\end{itemize}
An extended discussion from a philosophical perspective is in

\begin{itemize}%
\item [[Colin McLarty]], \emph{Exploring Categorical Foundations} , Phil. Math. \textbf{12} no.3 (2004) 37-53.

\end{itemize}
For a more recent review from a critical perspective containing additional recent references see

\begin{itemize}%
\item [[Solomon Feferman]], \emph{Foundations of Unlimited Category Theory: What Remains to be Done} , Review of Symbolic Logic \textbf{6} no.1 (2013) 6-15. (\href{http://math.stanford.edu/~feferman/papers/FCT-RSL-2013.pdf}{pdf})

\end{itemize}
An informative discussion of the pros and cons of Lawvere's approach can be found in

\begin{itemize}%
\item [[Jean-Pierre Marquis]], \emph{Kreisel and Lawvere on Category Theory and the Foundations of Mathematics} . (\href{http://www.math.mcgill.ca/rags/seminar/Marquis_KreiselLawvere.pdf}{pdf-slides})

\end{itemize}
Palmgren's ideas can be found here:

\begin{itemize}%
\item [[Erik Palmgren]], \emph{Constructivist and Structuralist Foundations: Bishop's and Lawvere's Theories of Sets} , Ann. Pure. App. Logic \textbf{163} no.10 (2012) 1384-1399. (\href{http://www.math.uu.se/~palmgren/cetcs.pdf}{pdf})

\end{itemize}
For the relation between the theory of well-pointed toposes and \textbf{weak Zermelo set theory} as elucidated by work of [[Julian Cole|J. Cole]], [[Barry Mitchell]], and [[Gerhard Osius|G. Osius]] in the early 1970s see

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Topos Theory} , Academic Press New York 1977 (Dover reprint 2014). (sections 9.2-3)


\item [[Barry Mitchell]], \emph{Boolean Topoi and the Theory of Sets} , JPAA \textbf{2} (1972) pp.261-274.


\item [[Gerhard Osius]], \emph{Categorical Set Theory: A Characterization of the Category of Sets}, JPAA \textbf{4} (1974) 79-119.



\end{itemize}
[[!redirects ETCS]] [[!redirects etcs]] [[!redirects elementary theory of the category of sets]]



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