#Contents# * automatic table of contents goes here {:toc} ## Adjunctions ### Adjoint pairs in $2$-categories #### Mates ### For quasicategories (In terms of cographs/correspondences) #### 1-categorical motivation Every [[profunctor]] $$ k : C^{op} \times D \to S $$ defines a category $C *^k D$ with $Obj(C *^k D) = Obj(C) \sqcup Obj(D)$ and $$ Hom_{C^{op} \times D}(X,Y) = \left\{ \array{ Hom_C(X,Y) & if X, Y \in C \\ Hom_{D}(X,Y) & if X,Y \in D \\ k(X,Y) & if X \in C and Y \in D \\ \emptyset & otherwise } \right. \,. $$ This category naturally comes with a functor to the [[interval]] category $$ C *^k D \to \Delta^1 \,. $$ Now, every functor $L : C \to D$ induces a [[profunctor]] $$ k_L(X,Y) = Hom_D(L(X), Y) $$ and every functor $R : D \to C$ induces a [[profunctor]] $$ k_R(X,Y) = Hom_C(X, R(Y)) \,. $$ The functors $L$ and $R$ are adjoint precisely if the [[profunctors]] that they define in the above way are equal. This in turn is the case if $C \star^L D \simeq (D^{op} \star^{R^{op}} C^{op})^{op}$. We say that $C \star^k D$ is the [[cograph of a functor|cograph of the functor]] $k$. See there for more on this. #### The Definition The above characterization of adjoint functors in terms of categories over the interval is used in section 5.2.2 of * [[Jacob Lurie]], _[[Higher Topos Theory]]_ (motivated from the discussion of correspondences in section 2.3.1) to give a definition of adjunction between [[(infinity,1)-functors]]. +-- {: .un_defn} ###### Definition Let $C$ and $D$ be [[quasi-category|quasi-categories]]. An **adjunction** between $C$ and $D$ is * a morphism of [[simplicial sets]] $K \to \Delta^1$ * which is * a [[Cartesian fibration]] * a [[coCartesian fibration]] * such that $C \simeq K_{0}$ and $D\simeq K_{1}$. =-- For more on this see * [[adjoint (∞,1)-functor]]. ## Monads, Modalities and Closures ### Example: Reflective subcategories #### The Problem of Subtoposes +-- {: .num_lemma} ###### Theorem ([[nLab:Elephant]] A.4.3.9, p.192) Let $(l\dashv i):L\stackrel{i}{\hookrightarrow}E$ be a reflective subcategory of a topos such that the monad $i\circ l$ is [[nLab:cartesian monad|cartesian]] (i.e. $i\circ l$ preserves pullbacks). Then $L$ is a topos and $l$ preserves finite limits (i.e. $(l\dashv i)$ is a geometric morphism). =-- The following lemma improves on the statement * *A reflective subcategory of a topos is a topos if the reflector is left exact.* +-- {: .num_lemma} ###### Lemma Let $(L\dashv R):E\stackrel{\R}{\hookrightarrow} H$ be a reflective subcategory of a topos. Then $E$ is a topos if $L$ preserves pullbacks in the image of $a_H\circ R_!$ where * $(a_H\dashv Y_H):H\to Psh(H)$ is the left adjoint of the Yoneda embedding of $H$. * $R_!:=Lan_{Y_E} Y_H\circ R$ is the left Kan extension of $Y_H\circ R$ along the Yoneda embedding of $E$. $$\array{ Psh(E)&\stackrel{a_E\dashv Y_E}{\to}&E \\ \downarrow^{R_!}&&\downarrow^{L\dashv R} \\ Psh(H)&\stackrel{a_H\dashv Y_H}{\to}&H }$$ =-- +-- {: .proof} ###### Proof 1. The Yoneda embeddings of $E$ and $H$ both posess left adjoints: $H$ and $E$ are [[total category|total]]: Since $H$ is a topos, $H$ is total, since $E$ is a reflective subcategory of a total category $E$ is total. By the adjoint functor theorem for total categories this implies that the Yoneda embeddings of $E$ and $H$ both posess left adjoints. 1. We have $a_E\simeq L\circ a_H\circ R_!$. If this composite is left exact it exhibits $E$ as a left exact localization of a category of presheaves and hence in this case $E$ is a topos. 1. $a_H\circ R_!$ sends colimits to limits, since $R_!$ (as every Yoneda extension) commutes with colimits and $a_H$ as a left adjoint sends colimits to limits. 1. Hence $a_E\simeq L\circ a_H\circ R_!$ is left exact iff $L$ preserves limits in the image of $a_H\circ R_!$. 1. Since a reflector always preserves terminal objects (and all finite limits can be constructed from pullbacks and the terminal object), it is sufficient to check if $L$ preserves pullbacks in the image of $a_H\circ R_!$. =-- +-- {: .num_theorem} ###### Theorem Every subcategory of a category of presheaves which is reflective and coreflective is itself a category of presheaves (this is quoted at [[nLab:reflective subcategory]] as Bashir Velebil). In particular $E$ is a topos.) =-- ### Relation to factorization systems, reflective subfibrations +-- {: .num_theorem} ###### Theorem (1) The following statements are equivalent: * $(E,M)$ is a *reflective factorization system* in $H$. * There is a reflective subcategory $C\hookrightarrow H$ with reflector $\sharp$, $E$ is the class of morphisms whose $\sharp$-image is invertible in $C$, and $C=M/1$. * $(E,M)$ is a factorization system and $E$ satisfies 2-out -of-3. * $(E,M)$ is a factorization system and $M$ is the class of fibrant morphisms $P\to A$ which as dependent types $x:A\dashv P(x): Type$ satisfy $forall\, x \,in Rsc(P(x))$. * For every $H$-morphism $f:A\to B$ satisfying: $\sharp A$ and $\sharp B$ are contractible, also for all $b$ we have $\sharp \, hFiber(f,b)$ is contractible. $$is Contr(\sharp A),is Contr(\sharp B), f:A\to B, b:B\vdash is Contr (\sharp h Fiber(f,b))$$ (2) The following statements are equivalent: * $(E,M)$ is a factorization system in $H$. * The class $(E,M)^\times:=\{M/x|x\in H,\M/x\hookrightarrow H/x\,is.refl,\,refl.fact\}$ is pullback-stable where $refl.fact$ means that each reflection is defined by $(E,M)$-factorization. * $(C.x\subseteq H/x)_{x\in H}$ is a pullback-stable system of reflective subcategories of slices of $H$, and for every $x$ the class of objects of $C.x$ is closed under composition. * The class of types $B$ satisfying $in Rsc (B)$ is closed under dependent sums. $$in Rsc(A), forall \, x,in Rsc(P(x))\vdash in Rsc (\sum_{x:A} P(x))$$ (3) The following statements are equivalent: * $(C.x\subseteq H/x)_{x\in H}$ is a pullback-stable system of reflective subcategories of slices of $H$, for every $x$ the class of objects of $C.x$ is closed under composition, and all reflectors commute with pullbacks. * The (by (2)) to $(C.x\subseteq H/x)_{x\in H}$ corresponding factorization system $(E,M)$ is pullback stable. =-- ### Module of a Monad ### Modalities ### Example: The Lawvere Tierney operator +-- {: .num_defn} ###### Definition A __Lawvere--Tierney topology__ in $E$ is ([[nLab:internalization|internally]]) a [[nLab:closure operator]] given by a [[nLab:exact functor|left exact]] [[nLab:idempotent monad]] on the internal meet-[[nLab:semilattice]] $\Omega$. This means that: a Lawvere--Tierney topology in $E$ is a [[nLab:morphism]] $$ j: \Omega \to \Omega $$ such that 1. $j true = true$, equivalently $\id_\Omega \leq j: \Omega \to \Omega$ ('if $p$ is true, then $p$ is locally true') $$ \array{ * &\stackrel{true}{\to}& \Omega \\ & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega } $$ 1. $j j = j$ (?$p$ is locally locally true iff $p$ is locally true'); $$ \array{ \Omega &\stackrel{j}{\to}& \Omega \\ & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega } $$ 1. $j \circ \wedge = \wedge \circ (j \times j)$ (?$p \wedge q$ is locally true iff $p$ and $q$ are each locally true') $$ \array{ \Omega \times \Omega &\stackrel{\wedge}{\to}& \Omega \\ {}^{\mathllap{j \times j}}\downarrow && \downarrow^{\mathrlap{j}} \\ \Omega \times \Omega &\underset{\wedge}{\to}& \Omega } \,. $$ =-- Here $\leq$ is the internal [[nLab:partial order]] on $\Omega$, and $\wedge: \Omega \times \Omega \to \Omega$ is the internal [[nLab:meet]]. This appears for instance as ([MacLaneMoerdijk, V 1.](#MacLaneMoerdijk)). ### Example: cohesive topos