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\begin{document}

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\section*{axiom of separation}

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

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

[[!include foundations - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{axiom_schemes_of_separation}{}\section*{{Axiom schemes of separation}}\label{axiom_schemes_of_separation}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statements}{Statements}\dotfill \pageref*{statements} \linebreak
\noindent\hyperlink{in_structural_set_theory}{In structural set theory}\dotfill \pageref*{in_structural_set_theory} \linebreak
\noindent\hyperlink{in_hyperdoctrines}{In hyperdoctrines}\dotfill \pageref*{in_hyperdoctrines} \linebreak
\noindent\hyperlink{LawvereDefinition}{Lawvere's definition}\dotfill \pageref*{LawvereDefinition} \linebreak
\noindent\hyperlink{ehrhards_reformulation}{Ehrhard's reformulation}\dotfill \pageref*{ehrhards_reformulation} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{subsets}{Subsets}\dotfill \pageref*{subsets} \linebreak
\noindent\hyperlink{presheaves}{Presheaves}\dotfill \pageref*{presheaves} \linebreak
\noindent\hyperlink{ExamplesDependentLinearTypeTheory}{In dependent linear type theory}\dotfill \pageref*{ExamplesDependentLinearTypeTheory} \linebreak
\noindent\hyperlink{the_comprehensive_factorization_system}{The comprehensive factorization system}\dotfill \pageref*{the_comprehensive_factorization_system} \linebreak
\noindent\hyperlink{axiom_or_axiom_scheme}{Axiom or axiom scheme?}\dotfill \pageref*{axiom_or_axiom_scheme} \linebreak
\noindent\hyperlink{relation_to_the_axiom_of_replacement}{Relation to the axiom of replacement}\dotfill \pageref*{relation_to_the_axiom_of_replacement} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_hyperdoctrines_2}{In hyperdoctrines}\dotfill \pageref*{in_hyperdoctrines_2} \linebreak
\noindent\hyperlink{in_linear_logic}{In linear logic}\dotfill \pageref*{in_linear_logic} \linebreak


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

In [[set theory]], the \textbf{axiom scheme of separation} aka \textbf{specification} states that, given any [[set]] $X$ and any [[property]] $P$ of the [[elements]] of $X$, there is a set

\begin{displaymath}
\{ X | P \} = \{ a \in X \;|\; P(a) \}
\end{displaymath}
consisting precisely of those elements of $X$ for which $P$ holds:

\begin{displaymath}
a \in \{ X | P \} \;\leftrightarrow\; a \in X \;\wedge\; P(a) .
\end{displaymath}
Note that $\{X|P\}$ is a [[subset]] of $X$.

It is important to specify what language $P$ can be written in. This connects the axiom to [[logic]] and the [[foundations of mathematics]]. Arguably, [[first-order logic]] developed in order to explain the meaning of [[Ernst Zermelo]]'s axiom of separation (although Zermelo himself disagreed with the interpretation that this gave). Separation is (usually) given as an axiom \emph{scheme} because there is one axiom for each way to state a property in the language. (We also allow parameters in $P$.)

\hypertarget{statements}{}\subsection*{{Statements}}\label{statements}

From weaker to stronger:

\begin{itemize}%
\item For \textbf{bounded separation} aka \textbf{restricted separation} aka \textbf{$\Delta_0$-separation}, $P$ must be written in the language of first-order set theory and all [[quantifications]] must be guarded by a set: of the form $\forall x \in A$ or $\exists x \in A$ for some set $A$.


\item For \textbf{limited separation} aka \textbf{$\Delta_0^{\mathcal{P}}$-separation}, $P$ must be written in the language of first-order set theory and all quantifications must be guarded by a set or a [[power class]]: of the form $\forall x \in A$, $\exists B \subseteq A$, etc. Limited separation trivially implies bounded separation, while bounded separation implies limited separation if [[power sets]] exist.


\item For \textbf{full separation} aka simply \textbf{separation}, $P$ must be written in the language of first-order set theory, but otherwise anything goes; in a [[class theory]], $P$ must be guarded by a class. Full separation trivially implies limited separation.


\item For \textbf{large separation}, $P$ must be written in the language of first-order class theory; of course, this only makes sense \emph{in} a class theory. The difference in strength between the class theories $MK$ and $NBG$ is precisely that the former has large separation but the latter does not.


\item Separation is sometimes called \textbf{restricted comprehension}; for \textbf{full comprehension} aka simply \textbf{comprehension}, no set $X$ needs to be given ahead of time. Full comprehension was proposed by [[Gottlob Frege]], but leads to [[Russell's paradox]]. However, full comprehension can sometimes be allowed if the ambient logic is nonclassical, such as [[linear logic]] or [[paraconsistent logic]].


\item For \textbf{stratified comprehension}, no set $X$ is given, but $P$ is restricted to stratified formulas, in which each [[variable]] $x$ can be given a consistent [[natural number]] $\sigma(x)$ (its stratification) such that $x \in y$ appears in the formula only if $\sigma(y) = \sigma(x) + 1$. This is used in [[Van Quine]]'s [[New Foundations]].



\end{itemize}
\hypertarget{in_structural_set_theory}{}\subsection*{{In structural set theory}}\label{in_structural_set_theory}

Set theory is usually given in [[material set theory|material]] form, with a language based on a global membership relation $\in$, and we have implicitly followed this above. However, separation makes sense also in [[structural set theory]] (although full comprehension does not, except in a structural class theory with a [[class of all sets]], where it again leads to paradox). The conclusion of the axiom is the existence of a set

\begin{displaymath}
\{ X | P \} = \{ a\colon X \;|\; P(a) \}
\end{displaymath}
and an [[injection]] $i_P\colon \{X|P\} \to X$ such that

\begin{displaymath}
\exists b\colon S,\; a = i_P(b) \;\leftrightarrow\; P(a) .
\end{displaymath}
Note that $\{X|P\}$, equipped with $i_P$, is a [[subset]] of $X$ in the structural sense.

The structural axioms can of course be stated even in a material set theory, where they are actually weaker than the corresponding material axioms; however, the material axioms follow (as usual) from the structural axioms using [[axiom of replacement|restricted replacement]], which is quite weak (and also follows from the material form of bounded separation).

If a structural set theory is given by stating axioms for the [[category of sets]], then restricted separation amounts to the property that this category is a [[Heyting category]]. If it is an [[elementary topos]], then since it satisfies the power set axiom, this implies limited separation as well. Full separation is somewhat less natural to state category-theoretically, but the combination of full separation with the structural [[axiom of collection]] is equivalent to saying that the category of sets is [[autological category|autological]].

\hypertarget{in_hyperdoctrines}{}\subsection*{{In hyperdoctrines}}\label{in_hyperdoctrines}

\hypertarget{LawvereDefinition}{}\subsubsection*{{Lawvere's definition}}\label{LawvereDefinition}

[[Lawvere]] gives a definition (\hyperlink{Lawvere70}{Lawvere70, p. 12-13}) of comprehension in [[hyperdoctrines]].

\begin{defn}
\label{LawvereianComprehension}\hypertarget{LawvereianComprehension}{}
Let $p \colon E \to B$ be a [[bifibration]] over the category $B$, and assume that each [[fibre]] $E_X$ of $E$ has a [[terminal object]] $T_X$. For any [[morphism]] $t \colon Y \to X$ in $B$, define the \emph{image} $im t$ of $t$ to be the pushforward ([[dependent sum]]) $t_! T_Y$. This gives rise to a functor $im \colon B/X \to E_X$ for each $X$. Then $E$ is said to \emph{have comprehension}, or to satisfy the \emph{comprehension schema}, if each image functor has a [[right adjoint]] $\{-\} \colon E_X \to B/X$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This means that for each $P \in E_X$ there is a morphism $i_P \colon \{ P \} \to X$ in $B$ such that there is a [[bijection]] between commuting triangles $t = i_P \circ t'$ in $B$ and morphisms $im t \to P$ in $E_X$. Lawvere calls the morphism $\{P\} \to X$ the \emph{[[extension (semantics)|extension]]} of $P$, so that one could say that $E$ satisfies the comprehension schema if the extensions of all predicates exist.

\end{remark}
\hypertarget{ehrhards_reformulation}{}\subsubsection*{{Ehrhard's reformulation}}\label{ehrhards_reformulation}

Notice that, in the above situation, morphisms $\im t \to P$ in $E_X$ are in bijection with morphisms $T_Y \to t^* P$ in $E_Y$, and that these are the same as morphisms $T_Y \to P$ in the total category $E$ that lie over $t$. This leads to an alternative formulation of the definition, originally considered by \hyperlink{Ehrhard88}{Ehrhard 1988} (see also \hyperlink{Jacobs93}{Jacobs 1993}, \hyperlink{Jacobs99}{Jacobs 99}), which does not depend on $p$ being a bifibration.

Suppose the fibres $E_X$ have terminal objects that are preserved by the reindexing functors. This is equivalent to saying that $p:E\to B$ has a section $T:B\to E$ taking each object $X$ to a terminal object $T_X$ of $E_X$ and each morphism to a cartesian arrow; such a functor is then automatically [[fully faithful]] and [[right adjoint]] to $p:E\to B$. If $B$ has a terminal object $1$, then this is additionally equivalent to saying that the entire category $E$ has a terminal object $T_1$ preserved by $p$, since then $X^*(T_1)$ will be terminal in $E_X$ by combining the universal properties of $X^*$ and $T_1$. Note also that if $p$ is a bifibration, as in Lawvere's definition, then the reindexing functors are right adjoints and hence automatically preserve all terminal objects.

Comprehension in the sense above is equivalent to the existence of a further right adjoint to the terminal-object functor $T:B\to E$.

The implication from \hyperlink{LawvereDefinition}{Lawvere's definition} to Ehrhard's is clear. Conversely, if a fibration $p \colon E \to B$ satisfies the Ehrhard definition, then there is a bijection between morphisms $T_Y \to P$ in $E$ and morphisms $Y \to \{P\}$ in $B$. One must then show that the composite of this last with the canonical $\{P\} \to X$ (given by $p$ applied to the counit $T_{\{P\}} \to P$) is equal to $p$ applied to the morphism $Y \to \{P\}$, thus giving a morphism in $B/X$ of the right sort. But in fact the morphism $Y \to \{P\}$ is unique with the property that applying the functor $T_{-}$ and composing with the counit gives the morphism $T_Y \to P$, by the counit's universal property, and applying $p$ to this commuting triangle in $E$ produces the required one in $B$.

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\hypertarget{subsets}{}\paragraph*{{Subsets}}\label{subsets}

The tautological example which is useful to see what the abstract \hyperlink{LawvereDefinition}{definition by Lawvere} axiomatizes is the following.

Let $B =$[[Set]] be the category of sets and for $X$ a [[set]] let $E_X$ be the [[poset]] of [[subsets]] of $X$, regarded as the [[propositions]] about elements in $X$. Then comprehension exists and is given by sending a subset of $X$ regarded as an object of $E_X$ (hence regarded as a proposition) to the same subset, but now regarded as a [[monomorphism]] in [[Set]] into $X$.

More generally, the same construction works for the posets of subobjects in any [[regular category]].

\hypertarget{presheaves}{}\paragraph*{{Presheaves}}\label{presheaves}

There is a functor $Cat_1^{op} \to CAT_1$, from the [[large category|large]] [[Cat|1-category of categories]] and functors to the `[[very large category|very large]]' 1-category of large categories and functors, that sends a category $C$ to the [[category of presheaves]] $[C^{op}, Set]$ on $C$, and a functor $F \colon C \to D$ to the pullback functor $F^* \colon [D^{op}, Set] \to [C^{op}, Set]$. These pullback functors have left and right adjoints given by [[Kan extension]].

\hyperlink{Lawvere70}{Lawvere 70} shows that this fibration has comprehension, with the [[extension (semantics)|extension]] of a presheaf given by its [[category of elements]] together with the canonical projection from this to the base category.

\hypertarget{ExamplesDependentLinearTypeTheory}{}\paragraph*{{In dependent linear type theory}}\label{ExamplesDependentLinearTypeTheory}

If the [[hyperdoctrine]] has [[linear type theories]]/[[symmetric monoidal categories]] as fibers, then it is more natural to consider in def. \ref{LawvereianComprehension} not the [[terminal object]] in some fiber (if any) but the [[tensor unit]] (which of course happens to be the terminal object in the case of [[cartesian monoidal category|cartesian monoidal]]) fibers.

In this case then the image functor in def. \ref{LawvereianComprehension} is

\begin{displaymath}
\Sigma_X \;\colon\; (Y \stackrel{f}{\to} X) \mapsto \underset{f}{\sum} 1_Y 
  \,.
\end{displaymath}
If this has a [[right adjoint]] $R_X$, hence if we have linear comprehension according to def. \ref{LawvereianComprehension}, then the induced [[comonad]]

\begin{displaymath}
!_X \coloneqq \Sigma_X \circ R_X
\end{displaymath}
is (the [[dependent type theory|dependent]] version of) the canonical [[categorical semantics|categorical interpretation]] of the [[exponential modality]] of [[linear logic]]. See at \emph{\href{http://ncatlab.org/nlab/show/dependent+linear+type+theory#TheCanonicalComodality}{dependent linear type theory -- The canonical co-modality}} for more.

\hypertarget{the_comprehensive_factorization_system}{}\subsubsection*{{The comprehensive factorization system}}\label{the_comprehensive_factorization_system}

If the extension functors $E_X \to B/X$ are [[fully faithful]], then together they make $E$ into a `fibrewise [[reflective subcategory|reflective]]' subfibration of the codomain fibration of $B$, which is a reflective subfibration if the image functors preserve pullbacks. (See [[stable factorization system]].)

Call a morphism of the form $\{P\} \to X$ a \emph{subtype inclusion}. Every morphism $t \colon Y \to X$ of $B$ factors through one such by means of the unit $\eta_t$ of the adjunction $im \dashv \{-\}$:

\begin{displaymath}
\itexarray{
  Y & & \overset{\eta_t}{\to} & & \{ im t \} \\
  & \mathllap{t}\searrow & & \swarrow\mathrlap{i_t} & \\
  & & X & &
}
\end{displaymath}
This gives rise to an [[orthogonal factorization system]] precisely when each $\eta_t$ is [[orthogonal]] to all subtype inclusions. It is shown by \hyperlink{Carboni97}{(Carboni et. al., section 2.12)} that this holds if and only if subtype inclusions are closed under composition.

For the subobject fibration of a regular category, this gives the usual ([[regular epimorphism|regular epi]], [[monomorphism|mono]]) factorization system, while for the fibration of presheaves over $Cat$ it gives the factorization of a functor into a [[final functor]] followed by a [[discrete fibration]]. (See also [[michaelshulman:comprehensive factorization]] for a description of the latter as a [[factorization system in a 2-category]].)

\hypertarget{axiom_or_axiom_scheme}{}\subsection*{{Axiom or axiom scheme?}}\label{axiom_or_axiom_scheme}

Although usually presented as an axiom scheme, in many cases, all instances of separation follow from finitely many special cases (which can then be packaged into a single axiom, using [[and|conjunction]], although this is probably pointless). This is the case, for example, in [[ETCS]] (a structural set theory that satisfied bounded separation) and [[NBG]] (a material class theory that satisfies full separation). In [[type theory|type-theoretic]] foundations of mathematics, separation is usually invisible, but again some form (generally only bounded) can again be proved from a few specific axioms or constructions.

\hypertarget{relation_to_the_axiom_of_replacement}{}\subsection*{{Relation to the axiom of replacement}}\label{relation_to_the_axiom_of_replacement}

Full separation follows from the [[axiom of replacement]] and the principle of [[excluded middle]] (along with the axiom of the [[empty set]]). Therefore, the axiom is often left out entirely of a description of [[ZFC]] (the usually accepted foundation of mathematics). In versions of set theory for [[constructive mathematics]], however, we often have replacement but only bounded or limited separation, and in any case separation must be listed explicitly.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

(\ldots{})

\hypertarget{in_hyperdoctrines_2}{}\subsubsection*{{In hyperdoctrines}}\label{in_hyperdoctrines_2}

\begin{itemize}%
\item [[Lawvere]], F. W. \emph{Equality in hyperdoctrines and comprehension schema as an adjoint functor}. In A. Heller, ed., \emph{Proc. New York Symp. on Applications of Categorical Algebra}, pp. 1--14. AMS, 1970. ([[LawvereComprehension.pdf:file]])

\end{itemize}
\begin{itemize}%
\item [[Lawvere]], F. W. \emph{Metric spaces, generalized logic and closed categories} Rend. Sem. Mat. Fis. Milano, 43:135--166 (1973). Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 (\href{http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html}{tac})

\end{itemize}
\begin{itemize}%
\item [[Thomas Ehrhard]]. A Categorical Semantics of Constructions. LICS 1988. (\href{https://www.irif.univ-paris-diderot.fr/~ehrhard/pub/categ-sem-constr.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Bart Jacobs]]. Comprehension categories and the semantics of type dependency. \emph{Theoretical Computer Science} 107:2 (1993), pp. 169-207.

\end{itemize}
\begin{itemize}%
\item [[Bart Jacobs]]. \emph{Categorical Logic and Type Theory}. Elsevier, 1999.

\end{itemize}
\begin{itemize}%
\item [[Finn Lawler]], section ``Tabulation and comprehension'' in \emph{[[finnlawler:thesis outline|Fibrations of predicates and bicategories of relations]]} (2014)


\item Carboni, Janelidze, Kelly, Par\'e{}. \emph{On localization and stabilization for factorization systems}. Applied Categorical Structures, 1997.



\end{itemize}
\hypertarget{in_linear_logic}{}\subsubsection*{{In linear logic}}\label{in_linear_logic}

Full comprehension (a.k.a. ``naive set theory'') in the context of logic without the [[contraction rule]] is discussed in:

\begin{itemize}%
\item Grishin, V. N., ``Predicate and set theoretic calculi based on logic without contraction rules'' (Russian), \emph{Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya}, 45(1): 47 -- 68, 1981. English translation in Math. USSR Izv., 18(1): 41 -- 59, 1982. (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1547&option_lang=eng}{math-net.ru})


\item [[Jean-Yves Girard]], Light Linear Logic, \emph{Information and Computation}, 14(3):123-137, 2003. (\href{http://iml.univ-mrs.fr/~girard/LLL.pdf.gz}{pdf.gz})


\item [[Kazushige Terui]], Light Affine Set Theory: A Naive Set Theory of Polynomial Time, \emph{Studia Logica: An International Journal for Symbolic Logic}, Vol. 77, No. 1 (Jun., 2004), pp. 9-40. (\href{http://www.jstor.org/stable/20016605}{jstor}) (\href{http://www.kurims.kyoto-u.ac.jp/~terui/lastfin.pdf}{pdf}). See also Terui's slides, \href{http://www.kurims.kyoto-u.ac.jp/~terui/summer3.pdf}{Linear Logic and Naive Set Theory (Make our garden grow)}



\end{itemize}
category: foundational axiom

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