\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{indexed monoidal (infinity,1)-category} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories} [[!include monoidal categories - contents]] \hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra} [[!include homotopy - contents]] \hypertarget{indexed_monoidal_categories}{}\section*{{Indexed monoidal $(\infty,1)$-categories}}\label{indexed_monoidal_categories} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{slices_of_a_topos}{Slices of a topos}\dotfill \pageref*{slices_of_a_topos} \linebreak \noindent\hyperlink{PointedObjects}{Parameterized pointed objects}\dotfill \pageref*{PointedObjects} \linebreak \noindent\hyperlink{ParameterizedModules}{Parameterized modules}\dotfill \pageref*{ParameterizedModules} \linebreak \noindent\hyperlink{ParameterizedModuleSpectra}{Parametrized module spectra}\dotfill \pageref*{ParameterizedModuleSpectra} \linebreak \noindent\hyperlink{parameterized_formal_moduli_problems}{Parameterized formal moduli problems}\dotfill \pageref*{parameterized_formal_moduli_problems} \linebreak \noindent\hyperlink{ForQuasicoherentSheaves}{Quasicoherent sheaves of modules}\dotfill \pageref*{ForQuasicoherentSheaves} \linebreak \noindent\hyperlink{stable_homotopy_theory_of_equivariant_spectra}{Stable homotopy theory of $G$-equivariant spectra}\dotfill \pageref*{stable_homotopy_theory_of_equivariant_spectra} \linebreak \noindent\hyperlink{Structures}{Structures in an indexed monoidal $(\infty,1)$-category}\dotfill \pageref*{Structures} \linebreak \noindent\hyperlink{TheCanonicalComodality}{Exponential modality and Fock spaces}\dotfill \pageref*{TheCanonicalComodality} \linebreak \noindent\hyperlink{dependent_linear_demorgan_duality}{Dependent linear deMorgan duality}\dotfill \pageref*{dependent_linear_demorgan_duality} \linebreak \noindent\hyperlink{PrimaryIntegralTransform}{Primary integral transforms}\dotfill \pageref*{PrimaryIntegralTransform} \linebreak \noindent\hyperlink{FundamentalClasses}{Fundamental classes}\dotfill \pageref*{FundamentalClasses} \linebreak \noindent\hyperlink{SecondaryIntegralTransforms}{Secondary integral transforms}\dotfill \pageref*{SecondaryIntegralTransforms} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} An \emph{indexed monoidal $(\infty,1)$-category} is the [[(∞,1)-category|(∞,1)-categorical]] version of an [[indexed monoidal category]]. That is, it consists of a ``base'' $(\infty,1)$-category $\mathcal{C}$ together with, for each $X\in \mathcal{C}$, a [[monoidal (∞,1)-category]] $Mod(X)$ varying functorially with $X$. In one of the fundamental examples, $\mathcal{C}$ is the $(\infty,1)$-category of [[∞-groupoids]] (``spaces''), while $Mod(X)$ is that of [[parametrized spectra]]. Indexed monoidal $(\infty,1)$-categories are conjectured to be the [[categorical semantics]] of [[linear homotopy type theory]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{SemanticsForLinearHomotopyTypeTheory}\hypertarget{SemanticsForLinearHomotopyTypeTheory}{} An \textbf{indexed monoidal $(\infty,1)$-category} is \begin{enumerate}% \item an [[(∞,1)-category]] $\mathcal{C}$ with [[finite (∞,1)-limits]]; \item an [[(∞,1)-functor]] $Mod \colon \mathcal{C}^{op} \to SymMonCat_\infty$ to [[symmetric monoidal (∞,1)-categories]]; \end{enumerate} such that \begin{enumerate}% \item each $Mod(X)$ is [[closed monoidal (infinity,1)-category|closed]] (with [[internal hom]] to be denoted $[-,-]$); \item for each $f \colon \Gamma_1 \to \Gamma_2$ in $Mor(\mathcal{C})$ the assigned [[(∞,1)-functor]] $f^\ast \colon Mod(\Gamma_2) \to Mod(\Gamma_1)$ has a [[left adjoint]] $f_!$ and a [[right adjoint]] $f_\ast$; \item The adjunction $(f_! \dashv f^\ast)$ satisfies [[Frobenius reciprocity]] and the [[Beck-Chevalley condition]]. \end{enumerate} \end{defn} In the conjectural syntax of dependent linear type theory, the objects of $\mathcal{C}$ correspond to [[contexts]], which is why we sometimes write them as $\Gamma$. Similarly, we may sometimes write $f_!$ and $f_\ast$ as $\sum_f$ and $\prod_f$ respectively. The statement of [[Frobenius reciprocity]] then is that \begin{displaymath} \underset{f}{\sum} \left( X \otimes f^\ast Y \right) \simeq \left( \underset{f}{\sum} X \right) \otimes Y \,. \end{displaymath} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} For each example we also spell out some of the abstract constructions (discussed in \emph{\hyperlink{Structures}{Structures}} below) realized in that model. We also include some examples of [[indexed monoidal category|indexed monoidal 1-categories]] (which are a special case). Although probably those examples should be moved to [[indexed monoidal category]]. \hypertarget{slices_of_a_topos}{}\subsubsection*{{Slices of a topos}}\label{slices_of_a_topos} \begin{example} \label{}\hypertarget{}{} For $\mathbf{H}$ a [[topos]], then its system $\mathbf{H}_{/(-)} \colon \mathbf{H}^{op} \to CartMonCat \to MonCat$ of [[slice toposes]] is an indexed monoidal category, hence an indexed monodial $(\infty,1)$-category. \end{example} \begin{remark} \label{}\hypertarget{}{} This example for dependent linear type theory is extremely ``non-linear''. For instance, it has almost no [[dualizable objects]]; the only one is the terminal object in each slice. Given that the formulas below for secondary integral transforms (def.\ref{SIT}) assume dualizable objects, this means that in this non-linear context there will be no nontrivial secondary integral transforms. \end{remark} We now pass gradually to more and more linear examples.x \hypertarget{PointedObjects}{}\subsubsection*{{Parameterized pointed objects}}\label{PointedObjects} A first step away from the Cartesian example above towards more genuinely linear types is the following. \begin{defn} \label{PointedObjectsInSlice}\hypertarget{PointedObjectsInSlice}{} Let $\mathbf{H}$ be a [[topos]]. For $X \in \mathbf{H}$ any [[object]], write \begin{displaymath} \mathcal{C}_X \coloneqq \mathbf{H}_{/X}^{X/} \end{displaymath} for the [[category of pointed objects]] in the [[slice topos]] $\mathbf{H}_{/X}$. Equipped with the [[smash product]] $\wedge_X$ this is a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}_X, \wedge_X, X \coprod X)$. \end{defn} \begin{prop} \label{}\hypertarget{}{} For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$, the [[base change]] [[inverse image]] $f^\ast$ restricts to a functor $f^\ast \colon \mathcal{C}_Y \longrightarrow \mathcal{C}_X$ and this makes \begin{displaymath} \mathbf{H}_{/(-)}^{(-)/} \;\colon\; \mathbf{H}^{op} \longrightarrow MonCat \end{displaymath} an indexed monoidal category. \end{prop} This appears as (\hyperlink{Shulman08}{Shulman 08, examples 12.13 and 13.7}) and (\hyperlink{Shulman12}{Shulman 12, example 2.33}). \begin{proof} For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$ then the [[base change]] [[inverse image]] $f^\ast \colon \mathbf{H}_{/Y} \longrightarrow \mathbf{H}_{/X}$ preserves pointedness, and the [[pushout]] functor $f_! \colon \mathbf{H}^{X/} \longrightarrow \mathbf{H}^{Y/}$ preserves co-pointedness. These two functors hence form an [[adjoint pair]] $(f_! \dashv f^\ast) \colon \mathcal{C}_X \longrightarrow \mathcal{C}_Y$. Moreover, since [[colimits]] in the under-over category $\mathbf{H}_{/X}^{X/}$ are computed as colimits in $\mathbf{H}$ of [[diagrams]] with an [[initial object]] adjoined, and since by the [[Giraud axioms]] in the [[topos]] $\mathbf{H}$ [[pullback]] preserves these colimits, it follows that $f^\ast \colon \mathcal{C}_Y \to \mathcal{C}_X$ preserves colimits. Finally by the discussion at \emph{[[category of pointed objects]]} we have that $\mathcal{C}_X$ and $\mathcal{C}_Y$ are [[locally presentable categories]], so that by the [[adjoint functor theorem]] it follows that $f^\ast$ has also a [[right adjoint]] $f_\ast \colon \mathcal{C}_X \to \mathcal{C}_Y$. To see that $f^\ast$ is a [[strong monoidal functor]] observe that the [[smash product]] is, by the discussion there, given by a [[pushout]] over [[coproducts]] and [[products]] in the [[slice topos]]. As above these are all preserved by [[pullback]]. Finally to see that $f^\ast$ is also a [[strong closed functor]] observe that the [[internal hom]] on [[pointed objects]] is, by the discussion there, a [[fiber product]] of cartesian internal homs. These are preserved by the above case, and the fiber product is preserved since $f^\ast$ preserves all limits. Hence $f^\ast$ preserves also the internal homs of pointed objects. \end{proof} \hypertarget{ParameterizedModules}{}\subsubsection*{{Parameterized modules}}\label{ParameterizedModules} The examples of genuinely linear objects in the sense of [[linear algebra]] are the following. \begin{prop} \label{IndexedMonoidalCategoryOfModuleBundles}\hypertarget{IndexedMonoidalCategoryOfModuleBundles}{} Let $E$ be a [[commutative ring]]. Write $E Mod$ for its [[category of modules]]. For $X\in$ [[Set]] write \begin{displaymath} E Mod(X) \coloneqq Func(X, E Mod) \simeq (E Mod)^{\times_{\vert X \vert}} \end{displaymath} for the [[functor category]] from $X$ (regarded as a [[category]]) to $E Mod$, hence for the [[bundles]] of $E$-[[modules]] over the [[discrete space]] $X$. Equipped with the [[tensor product of modules]] this is a [[symmetric monoidal category]] $E Mod^{\otimes}$. This construction yields a [[functor]] \begin{displaymath} E Mod \;\colon\; Set^{op} \longrightarrow SymMonCat \,. \end{displaymath} which is an indexed monoidal category. \end{prop} \begin{example} \label{MatrixAsIntegralKernel}\hypertarget{MatrixAsIntegralKernel}{} In the context of prop. \ref{IndexedMonoidalCategoryOfModuleBundles} consider $E = k$ a [[field]]. Then $k Mod \simeq Vect_k$ is the [[category]] [[Vect]] of $k$-[[vector spaces]]. For $X \in Set$ a set, an $X$-dependent object $A \in Mod(X)\simeq Vect_k(X)$ is just a collection of ${\vert X\vert}$ vector spaces $A_x$ for $x\in X$. For $X \in FinSet \hookrightarrow Set$ a [[finite set]], the [[dependent sum]] and [[dependent product]] operations coincide and produce the [[direct sum]] of vector spaces: \begin{displaymath} \underset{X}{\sum} A \simeq \underset{X}{\prod} A \simeq \underset{x\in X}{\oplus} A_x \in Vect_k(\ast) \,. \end{displaymath} Under this identification every morphism $f \in Mor(Set)$ with finite fibers carries a canonical untwisted fiberwise fundamental class, def. \ref{FiberwiseFundamentalClass}, $[f]_{canonical}$. For \begin{displaymath} \itexarray{ && X_1 \times X_2 \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X_1 && && X_2 } \end{displaymath} a product corrrespondence of [[finite sets]], then an [[integral kernel]] $K$ on this, according to def. \ref{IntegralKernel}, with $A_i = 1_{X_i}$, is equivalently an ${\vert X_1\vert}\times {\vert X_2\vert}$-array of elements in $k$, hence a [[matrix]] $K_{\bullet,\bullet}$. \end{example} \hypertarget{ParameterizedModuleSpectra}{}\subsubsection*{{Parametrized module spectra}}\label{ParameterizedModuleSpectra} The example of parameterized modules \hyperlink{ParameterizedModules}{above} has an evident generalization from [[linear algebra]] to [[stable homotopy theory]] with [[abelian categories|abelian]] [[categories of modules]] refined to [[stable (∞,1)-categories|stable]] [[(∞,1)-categories of ∞-modules]]. Despite what this [[higher category theory]]-terminology might make the reader feel, this refinement flows naturally along the same lines as the 1-categorical situation. One may view the axiomatics of an indexed monoidal $(\infty,1)$-category as neatly characterizing precisely this intimate similarity. \newline \begin{prop} \label{ParameterizedModuleSpectraAreAnIndexedMonoidalCategory}\hypertarget{ParameterizedModuleSpectraAreAnIndexedMonoidalCategory}{} Let $E$ be an [[E-∞ ring]] spectrum and write $E Mod$ for its [[(∞,1)-category of ∞-modules]]. For $X \in$ [[∞Grpd]] write \begin{displaymath} E Mod(X) \coloneqq Func(X,E Mod) \end{displaymath} for the [[(∞,1)-category of (∞,1)-functors]] from $X$ (regarded as an [[(∞,1)-category]]) to $E Mod$, hence for the [[parameterized spectra]] over $X$ with $E$-[[∞-module]] structure. Equipped with the [[smash product of spectra]] this is a [[symmetric monoidal (∞,1)-category]] $E Mod^\otimes$ This construction yields an [[(∞,1)-functor]] \begin{displaymath} E Mod \;\colon\; \infty Grpd^{op} \longrightarrow SymMonCat_\infty \end{displaymath} This is an indexed monoidal $(\infty,1)$-category. \end{prop} (\hyperlink{HopkinsLurie}{Hopkins-Lurie}, \hyperlink{Schreiber14}{Schreiber 14}) \begin{remark} \label{}\hypertarget{}{} In the case that $E = \mathbb{S}$ is the [[sphere spectrum]], then $\mathbb{S}Mod \simeq Spectra$ is just the plain [[(∞,1)-category of spectra]] and then the above is the theory of plain [[parameterized spectra]]. In this case the corresponding [[(∞,1)-Grothendieck construction]] is the [[tangent (∞,1)-topos]] $T(\infty Grpd)$ of [[∞Grpd]]. This happens to be itself an [[(∞,1)-topos]], and as such is conjecturally semantics for \emph{ordinary} [[homotopy type theory]]. If ordinary homotopy type theory is enhanced with rules and axioms motivated by such models, it also becomes able to speak about stable homotopy theory and parametrized stable objects; the relation between this theory and [[dependent linear type theory]] remains to be explored. \end{remark} \begin{prop} \label{}\hypertarget{}{} The $\Sigma$-functor, def. \ref{Sigma}, in the model of parameterized $E$-module spectra, prop. \ref{ParameterizedModuleSpectraAreAnIndexedMonoidalCategory}, is the [[suspension spectrum]] construction $\Sigma^\infty$ followed by [[smash product]] of spectra with $E$: \begin{displaymath} \Sigma(X) \simeq \Sigma^\infty(X)\wedge E \,. \end{displaymath} This is the $E$-[[generalized homology]]-spectrum of the [[∞-groupoid]] $X$. This construction does have a [[right adjoint]] $\Omega^\infty$, where $(\Sigma^\infty \dashv \Omega^\infty)$ is the $E$-[[stabilization]]-adjunction for [[∞Grpd]]. Hence [[parameterized spectra]] have an [[exponential modality]], def. \ref{SemanticsForExponentialModality}. \end{prop} \begin{prop} \label{}\hypertarget{}{} In the class of models of prop. \ref{ParameterizedModuleSpectraAreAnIndexedMonoidalCategory}, an indexed monoidal $(\infty,1)$-category encodes the theory of [[twisted generalized cohomology]]. [[!include twisted generalized cohomology in linear homotopy type theory -- table]] \end{prop} \hypertarget{parameterized_formal_moduli_problems}{}\subsubsection*{{Parameterized formal moduli problems}}\label{parameterized_formal_moduli_problems} \begin{defn} \label{ParameterizedFormalModuliProblems}\hypertarget{ParameterizedFormalModuliProblems}{} For $\mathbf{H}$ a [[differential cohesive (∞,1)-topos]] with [[infinitesimal shape modality]] $\Pi_{inf}$ write for any object $X\in \mathbf{H}$ \begin{displaymath} Mod(X) \hookrightarrow \mathbf{H}_{/X}^{X/} \end{displaymath} for the [[full sub-(∞,1)-category]] of that of pointed objects over $X$, def. \ref{PointedObjectsInSlice}, on those that are in the kernel of $\Pi_{inf}$. \end{defn} \begin{remark} \label{}\hypertarget{}{} The construction in \ref{ParameterizedFormalModuliProblems} has the interpretation as the category of generalized [[formal moduli problems]] parameterized over $X$. The genuine formal moduli problems satisfy one an extra exactness property of the kind discussed at \emph{[[cohesive (∞,1)-presheaf on E-∞ rings]]}. \end{remark} \begin{prop} \label{}\hypertarget{}{} Parameterized formal moduli problems as in def. \ref{ParameterizedFormalModuliProblems} conjecturally form semantics for non-unital linear homotopy-type theory. \end{prop} \hypertarget{ForQuasicoherentSheaves}{}\subsubsection*{{Quasicoherent sheaves of modules}}\label{ForQuasicoherentSheaves} Pull-push of [[quasicoherent sheaves]] is usually discussed as a [[Grothendieck context]] of [[six operations]], but under some conditions it also becomes a [[Wirthmüller context]] and hence an indexed monoidal $(\infty,1)$-category Using results of Lurie this follows in the full generality of [[E-∞ geometry]] ([[spectral geometry]]). Consider quasi-compact and quasi-separated [[E-∞ scheme|E-∞ algebraic spaces]] ([[spectral geometry|spectral algebraic spaces]]). (This includes precisely those [[spectral Deligne-Mumford stacks]] which have a [[scallop decomposition]], see \href{derived+Deligne-Mumford+stack#RelationToDerivedAlgebraicSpaces}{here}.) If $f \;\colon\; X \longrightarrow Y$ is a map between these which is \begin{enumerate}% \item locally almost of finite presentation; \item strongly proper; \item has finite [[Tor-amplitude]] \end{enumerate} then the left adjoint to pullback of [[quasicoherent sheaves]] exists \begin{displaymath} (f_! \dashv f^\ast) \;\colon\; QCoh(X) \stackrel{\overset{f_!}{\longrightarrow}}{\underset{f^\ast}{\longleftarrow}} QCoh(Y) \,. \end{displaymath} ([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 3.3.23]]) If $f$ is \begin{itemize}% \item quasi-affine \end{itemize} then the right adjoint exists \begin{displaymath} (f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}} QCoh(Y) \,. \end{displaymath} ([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|LurieQC, prop. 2.5.12]], [[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 2.5.12]]) The [[projection formula]] in the dual form \begin{displaymath} f_\ast A \otimes B \longrightarrow f_\ast (A\otimes f^\ast B) \end{displaymath} for $f$ quasi-compact and quasi-separated appears as ([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, remark 1.3.14]]). Now if all the conditions on $f$ hold, so that $(f_! \dashv f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \longrightarrow QCoh(Y)$, then passing to opposite categories $QCoh(X)^{op} \longrightarrow QCoh(Y)^{op}$ exchanges the roles of $f_!$ and $f_\ast$, makes the projection formula be as in the above discussion and hence yields a Wirthm\"u{}ller context. The existence of [[dualizing modules]] $K$ \begin{displaymath} D X = [X,K] \end{displaymath} is discussed in ([[Representability theorems|Lurie, Representability theorems, section 4.2]].) \hypertarget{stable_homotopy_theory_of_equivariant_spectra}{}\subsubsection*{{Stable homotopy theory of $G$-equivariant spectra}}\label{stable_homotopy_theory_of_equivariant_spectra} For a finite group, $G$, consider the spectral [[Mackey functors]] on the category of finite $G$-sets. These are [[G-spectrum|genuine G-spectra]] (see \href{equivariant+stable+homotopy+theory#in_terms_of_mackeyfunctors}{equivariant stable homotopy theory}) which together constitute a $G$-equivariant $\infty$-category (or $G$-$\infty$-category, see [[Parametrized Higher Category Theory and Higher Algebra]]). This sends any orbit $G/H$, for $H$ a subgroup of $G$, to the monoidal $\infty$-category of genuine $H$-spectra. This forms an indexend monoidal $(\infty,1)$-category, as do other instances of $(\infty, 1)$-categories of $T$-spectra, for $T$ an \href{Parametrized+Higher+Category+Theory+and+Higher+Algebra#AtomicOrbital}{orbital} $\infty$-category. \hypertarget{Structures}{}\subsection*{{Structures in an indexed monoidal $(\infty,1)$-category}}\label{Structures} We discuss here structures (constructions) that may be defined and studied within an indexed monoidal $(\infty,1)$-category. \hypertarget{TheCanonicalComodality}{}\subsubsection*{{Exponential modality and Fock spaces}}\label{TheCanonicalComodality} The original axiomatics for [[linear type theory]] in (\hyperlink{Girard87}{Girard 87}) contain in addition to the structures corresponding to a ([[star-autonomous category|star-autonomous]]) [[symmetric monoidal category|symmetric]] [[closed monoidal category]] a certain (co-)[[modality]] traditionally denoted ``$!$'', the \emph{[[exponential modality]]}. In (\hyperlink{Benton95}{Benton 95, p.9,15}, \hyperlink{Bierman95}{Bierman 95}) it is noticed (reviewed in (\hyperlink{Barber97}{Barber 97, p. 21 (26)})) that a natural [[categorical semantics]] for this modality identifies it with the [[comonad]] that is induced from a [[strong monoidal adjunction]] \begin{displaymath} (\Sigma \dashv \Omega) \;\colon\; Mod(\ast) \stackrel{\overset{\Sigma}{\leftarrow}}{\underset{\Omega}{\longrightarrow}} \mathcal{C} \end{displaymath} between the [[closed monoidal category|closed]] [[symmetric monoidal category]] $Mod(\ast)$ which interprets the given [[linear type theory]] and a [[cartesian monoidal category]] $\mathcal{C}$. If there is only the [[strong monoidal functor]] $\Sigma \;\colon\; \mathcal{C} \longrightarrow Mod(\ast)$ without possibly a [[right adjoint]] $\Omega$, then (\hyperlink{Barber97}{Barber 97, p. 21 (27)}) speaks of the \emph{structural [[fragment]]} of [[linear type theory]]. In (\hyperlink{PontoShulman12}{Ponto-Shulman 12}) it is observed that this in turn is canonically induced if $Mod(\ast)$ is the [[linear type theory]] over the trivial context $\ast$ of a dependent linear type theory ([[indexed closed monoidal category]]) with category of contexts being $\mathcal{C}$: \begin{defn} \label{Sigma}\hypertarget{Sigma}{} Let $Mod \colon \mathcal{C}^{op} \to MonCat$ be an indexed monoidal $(\infty,1)$-category. Then for $X \in \mathcal{C}$ an object, set \begin{displaymath} \Sigma(X) \coloneqq \underset{X}{\sum} 1_X \in Mod(\ast) \end{displaymath} and for $f \;\colon\; Y \longrightarrow X$ a [[morphism]] in $\mathcal{C}$ set \begin{displaymath} \Sigma(f) \coloneqq \Sigma(Y) = \underset{Y}{\sum} 1_Y \stackrel{\simeq}{\to} \underset{X}{\sum} \underset{f}{\sum} f^\ast 1_X \stackrel{\underset{X}{\sum}(\epsilon_f)}{\longrightarrow} \underset{X}{\sum} 1_X = \Sigma(X) \,. \end{displaymath} \end{defn} (In the typical kind of model this means to assign to a [[space]] $X$ the linear space of [[sections]] of the trivial [[line bundle]] over it.) \begin{prop} \label{SigmaFunctor}\hypertarget{SigmaFunctor}{} The construction in def. \ref{Sigma} gives a [[strong monoidal functor]] \begin{displaymath} \Sigma \;\colon\; \mathcal{C} \longrightarrow Mod(\ast) \end{displaymath} \end{prop} This is (\hyperlink{PontoShulman12}{Ponto-Shulman 12, (4.3)}). \begin{remark} \label{}\hypertarget{}{} Below in example \ref{SigmaFunctorAsSecondaryTransform} we see that the functor $\Sigma$ in prop. \ref{SigmaFunctor} is a special case of a general construction of secondary integral transforms in an indexed monoidal $(\infty,1)$-category. \end{remark} This suggests the following definition. \begin{defn} \label{SemanticsForExponentialModality}\hypertarget{SemanticsForExponentialModality}{} Given an indexed monoidal $(\infty,1)$-category such that the functor $\Sigma$ from prop. \ref{SigmaFunctor} has a strong monoidal [[right adjoint]] \begin{displaymath} \Omega \colon Mod(\ast) \longrightarrow \mathcal{C} \end{displaymath} we refer to the [[comonad]] of the $(\Sigma \dashv \Omega)$-[[adjunction]] \begin{displaymath} ! \coloneqq \Sigma \circ \Omega \colon Mod(\ast) \to Mod(\ast) \,. \end{displaymath} as the \emph{[[exponential modality]]} in the conjectural [[linear type theory]] over the point whose semantics is $Mod(\ast)$. \end{defn} \begin{remark} \label{}\hypertarget{}{} The condition in def. \ref{SemanticsForExponentialModality} that $\Sigma$ (and its relative/dependent versions) has a [[right adjoint]] $\Omega$ may also be understood as saying that the dependent linear type theory satisfies the \emph{[[axiom of comprehension]]}. See at \emph{\href{http://ncatlab.org/nlab/show/axiom+of+separation#ExamplesDependentLinearTypeTheory}{comprehension -- Examples -- In dependent linear type theory}} for more. \end{remark} \hypertarget{dependent_linear_demorgan_duality}{}\subsubsection*{{Dependent linear deMorgan duality}}\label{dependent_linear_demorgan_duality} \begin{defn} \label{LinearNegation}\hypertarget{LinearNegation}{} For $\mathcal{C}^\otimes$ a [[closed monoidal category]] with [[unit object]] $1$ and [[internal hom]] $[-,-]$ write \begin{displaymath} \mathbb{D} \coloneqq [-,1] \end{displaymath} for the [[functor]] given by internal hom into the unit object. Syntactically this corresponds to the \emph{linear [[negation]]} operation. \end{defn} \begin{prop} \label{DependentLinearDeMorganDuality}\hypertarget{DependentLinearDeMorganDuality}{} \textbf{(dependent linear de Morgan duality)} In an indexed monoidal $(\infty,1)$-category, linear negation, def. \ref{LinearNegation}, intertwines dependent sum and dependent product: \begin{displaymath} \underset{f}{\prod} \mathbb{D} \simeq \mathbb{D} \underset{f}{\sum} \end{displaymath} \end{prop} For proof see \href{Wirthmüller%20context#ComparisonOfPushForwardsAndWirthmuelleriso}{here} at \emph{[[Wirthmüller context]]}. \hypertarget{PrimaryIntegralTransform}{}\subsubsection*{{Primary integral transforms}}\label{PrimaryIntegralTransform} \begin{defn} \label{PolynomialFunctorsAndCorrespondences}\hypertarget{PolynomialFunctorsAndCorrespondences}{} Given an indexed monoidal $(\infty,1)$-category $Mod\colon \mathcal{C}^{op}\to SymMonCat$, and given objects $X_1, X_2$ of $\mathcal{C}$ then a \textbf{linear [[polynomial functor]]} \begin{displaymath} P \colon Mod(X_1) \to Mod(X_2) \end{displaymath} is a functor of the form \begin{displaymath} P \simeq \underset{f_2}{\sum} \underset{g}{\prod} f_1^\ast \end{displaymath} for a [[diagram]] in $\mathcal{C}$ of the form \begin{displaymath} \itexarray{ && Y &\stackrel{g}{\longrightarrow}& Z \\ & {}^{\mathllap{f_1}}\swarrow & && & \searrow^{\mathrlap{f_2}} \\ X_1 && && && X_2 } \,. \end{displaymath} If here $g = id$ then this diagram is a [[correspondence]] \begin{displaymath} \itexarray{ && Z \\ & {}^{\mathllap{f_1}}\swarrow & & \searrow^{\mathrlap{f_2}} \\ X_1 && && X_2 } \end{displaymath} and the resulting $P \simeq \underset{f_2}{\sum}f_1 ^\ast$ is called a \textbf{linear polynomial functor} or \textbf{primary [[integral transform]]}. \end{defn} \begin{prop} \label{}\hypertarget{}{} Given a [[correspondence]] as above, then the primary integral transform through it is equivalent to the pull-tensor-push operation through the [[product]] \begin{displaymath} \underset{f_2}{\sum} \circ f_1^\ast \simeq \underset{p_2}{\sum} \circ \left( \left(f_1,f_2\right)_!\left(1_Z\right) \otimes \left(-\right)\right) \circ p_1^\ast \,. \end{displaymath} \end{prop} \begin{proof} By [[Frobenius reciprocity]]. \end{proof} \hypertarget{FundamentalClasses}{}\subsubsection*{{Fundamental classes}}\label{FundamentalClasses} \begin{defn} \label{FiberwiseFundamentalClass}\hypertarget{FiberwiseFundamentalClass}{} A \emph{fiberwise twisted fundamental class} $[f]$ on a morphism $f \colon X\to Y$ in $\mathcal{C}$ is \begin{enumerate}% \item a choice of dualizable object $\tau \in Mod(Y)$ (the \emph{twist}); \item a choice of equivalence \begin{displaymath} \underset{f}{\sum} f^\ast 1_X \stackrel{\simeq}{\longrightarrow} \underset{f}{\prod} f^\ast \tau \,. \end{displaymath} \end{enumerate} \end{defn} In this form this is stated in (\hyperlink{Schreiber14}{Schreiber 14}). \begin{remark} \label{FundamentalClassFromAmbidexterity}\hypertarget{FundamentalClassFromAmbidexterity}{} A special case of def. \ref{FiberwiseFundamentalClass} is obtained when the twist vanishes, $\tau = 1$, and when dependent sum and dependent product are naturally equivalent for all objects, $\sum_f \simeq \prod_f$. In this case the two adjoints to $f^\ast$ coincide to form an [[ambidextrous adjunction]]. This case is considered in (\hyperlink{HopkinsLurie}{Hopkins-Lurie}). \end{remark} \begin{remark} \label{}\hypertarget{}{} In view of dependent linear de Morgan duality, prop. \ref{DependentLinearDeMorganDuality}, a fiberwise twisted fundamental class in def. \ref{FiberwiseFundamentalClass} is equivalently a choice of equivalence \begin{displaymath} \underset{f}{\sum} f^\ast 1_X \stackrel{\simeq}{\longrightarrow} \mathbb{D}\underset{f}{\sum} f^\ast \mathbb{D}\tau \,, \end{displaymath} hence an identification of the dependent sum of the unit with the dual of the dependent sum of the twist. \end{remark} \begin{prop} \label{WirthmullerIsomorphism}\hypertarget{WirthmullerIsomorphism}{} A twisted fiberwise fundamental class $[f]$, def. \ref{FiberwiseFundamentalClass}, induces for all dualizable $A\in Mod(X)$ a [[natural equivalence]] \begin{displaymath} \mathbb{D} \underset{f}{\sum} f^\ast \mathbb{D}(A\otimes \tau) \simeq \underset{f}{\sum}f^\ast A \,. \end{displaymath} \end{prop} This is the ``\href{Wirthmüller%20context#ComparisonOfPushForwardsAndWirthmuelleriso}{Wirthm\"u{}ller isomorphism}''. \begin{defn} \label{ComponentOfFiberwiseClass}\hypertarget{ComponentOfFiberwiseClass}{} For $[f]$ a twisted fiberwise fundamental class and for $A$ dualizable, write \begin{displaymath} [f]_A \;\colon\; A \otimes \tau \stackrel{\mathbb{D}_{\epsilon_{\mathbb{D}(A \otimes \tau)}}}{\longrightarrow} \mathbb{D} \underset{f}{\sum} f^\ast (\mathbb{D}(A\otimes \tau)) \stackrel{\simeq}{\longrightarrow} \underset{f}{\sum}f^\ast A \end{displaymath} for the composite of the linear dual of the $(\sum_f \dashv f^\ast)$-[[counit of an adjunction|adjunction counit]] and the Wirthm\"u{}ller isomorphism, prop. \ref{WirthmullerIsomorphism}. \end{defn} \begin{remark} \label{}\hypertarget{}{} The key point is that the morphism in def. \ref{ComponentOfFiberwiseClass} goes in the reverse direction of the [[counit of an adjunction|adjunction counit]]. In this way it plays a role in the construction of secondary integral transforms \hyperlink{SecondaryIntegralTransforms}{below}. \end{remark} \hypertarget{SecondaryIntegralTransforms}{}\subsubsection*{{Secondary integral transforms}}\label{SecondaryIntegralTransforms} \begin{defn} \label{IntegralKernel}\hypertarget{IntegralKernel}{} For \begin{displaymath} \itexarray{ && Z \\ & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}} \\ X_1 && && X_2 } \end{displaymath} a [[correspondence]] as in def. \ref{PolynomialFunctorsAndCorrespondences}, then an \emph{[[integral kernel]]} for it is the choice of \begin{enumerate}% \item two dualizable objects $A_i \in Mod(X_i)$; \item a morphism between their pullbacks to the correspondence space: \begin{displaymath} f_1^\ast A_1 \stackrel{K}{\longleftarrow} f_2^\ast A_2 \,. \end{displaymath} \end{enumerate} \end{defn} \begin{defn} \label{SIT}\hypertarget{SIT}{} Given \begin{enumerate}% \item a [[correspondence]] $X_1 \stackrel{f_1}{\longleftarrow} Z \stackrel{f_2}{\longrightarrow} X_2$ as in def. \ref{PolynomialFunctorsAndCorrespondences}; \item an [[integral kernel]] $K$ as in def. \ref{IntegralKernel}; \item a $\tau$-twisted fundamental class $[f_2]$ as in def. \ref{FiberwiseFundamentalClass} \end{enumerate} we say that the induced \emph{secondary integral transform} is the morphism \begin{displaymath} \int_Z \Xi d \mu_f \;\colon\; \mathbb{D} \underset{X_1}{\sum} A_1 \longrightarrow \mathbb{D}\underset{X_2}{\sum} (A_2\otimes \tau) \end{displaymath} which is the dual (the image under $\mathbb{D}(-)$) of the following composite: \begin{displaymath} \underset{X_1}{\sum} A_1 \stackrel{\underset{X_1}{\sum}\epsilon_{A_1}}{\longleftarrow} \underset{X_1}{\sum} \underset{f_1}{\sum} f_1^\ast A_1 \stackrel{\simeq}{\leftarrow} \underset{Z}{\sum}f_1^\ast A_1 \stackrel{\underset{Z}{\sum} K}{\longleftarrow} \underset{Z}{\sum}f_2^\ast A_2 \stackrel{\simeq}{\leftarrow} \underset{X_2}{\sum}\underset{f_2}{\sum} f_2^\ast A_2 \stackrel{\underset{X_2}{\sum}[f_2]_{A_2}}{\longleftarrow} \underset{X_2}{\sum} (A_2\otimes \tau) \,, \end{displaymath} where the morphism on the left is the [[counit of an adjunction|adjunction counit]], the morphism in the middle is the given [[integral kernel]], and the morphism on the right is the given fundamental class. \end{defn} In this form this appears in (\hyperlink{Schreiber14}{Schreiber 14}). Specialized to the ambidextrous case of remark \ref{FundamentalClassFromAmbidexterity}, this is equivalent to the construction in (\hyperlink{HopkinsLurie}{Hopkins-Lurie}). \begin{remark} \label{}\hypertarget{}{} A [[correspondence]] $X_1 \stackrel{f_1}{\longleftarrow} Z \stackrel{f_2}{\longrightarrow} X_2$ may be thought of as a space $Z$ of ``paths'' or ``trajectories'' that connect points in $X_2$ to points in $X_1$. From this point of view definition \ref{SIT} is a linear map that takes functions on $X_2$ to functions on $X_1$ by pointwise forming a sum over paths connecting these points and adding all the contributions of the given [[integral kernel]] along these paths. This is conceptually just how the [[path integral]] in [[physics]] is supposed to work, only that mostly it doesn't, due to lack of a proper definition. However, at least some path integrals for [[topological field theories]] may be realized as secondary integral transforms of the above kind. Notice that while in [[modal type theory]] the ([[comonad|co-]])[[monads]] $(f^\ast \sum_f \dashv f^\ast \prod_f)$ are pronounced as \emph{[[possibility]]} and \emph{[[necessity]]}, the monad $\prod_f f^\ast$ appearing above, via def. \ref{FiberwiseFundamentalClass}, may be pronounced \emph{randomness}, see at \emph{[[function monad]]} for more. \end{remark} \begin{example} \label{SigmaFunctorAsSecondaryTransform}\hypertarget{SigmaFunctorAsSecondaryTransform}{} Consider the special case def. \ref{SIT} where the right leg of the correspondence is the identity \begin{displaymath} \itexarray{ && Z \\ & {}^{\mathllap{f}}\swarrow && \searrow^{\mathrlap{=}} \\ X && && Z } \end{displaymath} and where the integral kernel is the identity \begin{displaymath} K = id_{1_Z} \colon id_Z^\ast 1_Z = 1_Z \to 1_Z = f^\ast 1_X \,. \end{displaymath} Then the associated secondary integral transform, def. \ref{SIT}, is the morphism \begin{displaymath} \underset{X}{\sum} 1_X \stackrel{\underset{\epsilon_f}{\sum}}{\longleftarrow} \underset{X}{\sum} \underset{f}{\sum} f^\ast 1_X \stackrel{\simeq}{\longleftarrow} \underset{Z}{\sum} 1_Z \,. \end{displaymath} This is just the $\Sigma$-functor of def. \ref{Sigma}: \begin{displaymath} \mathbb{D}\int_Z id \, d\mu_f = \Sigma(f) \; \colon \; \Sigma(Z) \stackrel{\Sigma(f)}{\longleftarrow} \Sigma(X) \,. \end{displaymath} \end{example} \begin{prop} \label{}\hypertarget{}{} Given a correspondence with integral kernel as in example \ref{MatrixAsIntegralKernel}, then the induced secondary integral transform according to def. \ref{SIT} is the [[linear function]] \begin{displaymath} k^{\vert X_1\vert} \longleftarrow k^{\vert X_2\vert} \end{displaymath} represented by that matrix. \end{prop} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[type theory]], [[homotopy type theory]] \item [[modal type theory]] \item [[stable homotopy type]] \item [[geometric homotopy type theory]] \item [[cohesive homotopy type theory]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} Plain [[linear type theory]] originates in \begin{itemize}% \item [[Jean-Yves Girard]], \emph{Linear logic}, Theoretical Computer Science 50:1, 1987. (\href{http://iml.univ-mrs.fr/~girard/linear.pdf}{pdf}) \end{itemize} The [[categorical semantics|categorical interpretation]] of Girard's $!$-[[modality]] as a [[comonad]] is due to \begin{itemize}% \item P. N. Benton, G. M. Bierman, [[Martin Hyland]], [[Valeria de Paiva]], \emph{Term assignment for intuitonistic linear logic}, Technial Report 262, Computer Laboratory, University of Cambridge, August 1992. \end{itemize} and that this is naturally to be thought of as arising from a [[monoidal adjunction]] between the closed [[symmetric monoidal category]] and a [[cartesian closed category]] is due to \begin{itemize}% \item G. Bierman, \emph{On Intuitionistic Linear Logic} PhD thesis, Computing Laboratory, University of Cambridge, 1995 (\href{http://research.microsoft.com/~gmb/papers/thesis.pdf}{pdf}) \item N. Benton, \emph{A mixed linear and non-linear logic; proofs, terms and models}, In \emph{Proceedings of Computer Science Logic} `94, vol. 933 of Lecture Notes in Computer Science. Verlag, June 1995. ([[BentonLinearLogic.pdf:file]]) \end{itemize} A review of all this and further discussion is in \begin{itemize}% \item Andrew Graham Barber, \emph{Linear Type Theories, Semantics and Action Calculi}, 1997 (\href{http://www.lfcs.inf.ed.ac.uk/reports/97/ECS-LFCS-97-371/‎}{web}, \href{http://www.lfcs.inf.ed.ac.uk/reports/97/ECS-LFCS-97-371/ECS-LFCS-97-371.pdf}{pdf}) \end{itemize} The 1-categorical vesrion of [[indexed closed monoidal categories]] is discussed in \begin{itemize}% \item [[Mike Shulman]], \emph{Framed bicategories and monoidal fibrations}, in Theory and Applications of Categories, Vol. 20, 2008, No. 18, pp 650-738. (\href{http://arxiv.org/abs/0706.1286}{arXiv:0706.1286}, \href{http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html}{TAC}) \item [[Kate Ponto]], [[Mike Shulman]], \emph{Duality and traces for indexed monoidal categories}, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 (\href{http://arxiv.org/abs/1211.1555}{arXiv:1211.1555}) \item [[Mike Shulman]], \emph{Enriched indexed categories} (\href{http://arxiv.org/abs/1212.3914}{arXiv:1212.3914}) \end{itemize} Comments on the formalization of secondary [[integral transforms]] and [[path integral]] [[quantization]] in an indexed monoidal $(\infty,1)$-category are in \begin{itemize}% \item [[Urs Schreiber]] \emph{[[schreiber:Quantization via Linear homotopy types]]} (\href{http://arxiv.org/abs/1402.7041}{arXiv:1402.7041}) \item [[Michael Hopkins]], [[Jacob Lurie]], \emph{[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]} \end{itemize} [[!redirects indexed monoidal (infinity,1)-categories]] [[!redirects indexed monoidal (∞,1)-category]] [[!redirects indexed monoidal (∞,1)-categories]] \end{document}