Spahn adjunctions, modalities and closures (Rev #1)

Adjunctions
- Adjoint pairs in $2$ -categories
  - Mates
- For quasicategories (In terms of cographs/correspondences)
  - 1-categorical motivation
  - The Definition
Monads, Modalities and Closures

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 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?.

Definition

Let $C$ and $D$ be 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

Theorem (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 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.

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

The Yoneda embeddings of $E$ and $H$ both posess left adjoints: $H$ and $E$ are 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.
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.
$a_H\circ R_!$ sends colimits to limits, since $R_!$ (as every Yoneda extension) commutes with colimits and $a_H$ as a left adjoint sends colimits to limits.
Hence $a_E\simeq L\circ a_H\circ R_!$ is left exact iff $L$ preserves limits in the image of $a_H\circ R_!$ .
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 $a_H\circ R_!$ .

Theorem

Every subcategory of a category of presheaves which is reflective and coreflective is itself a category of presheaves (this is quoted at reflective subcategory as Bashir Velebil). In particular $E$ is a topos.)

Relation to factorization systems, reflective subfibrations

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

Definition

A Lawvere–Tierney topology in $E$ is (internally) a closure operator given by a left exact idempotent monad on the internal meet-semilattice $\Omega$ .

This means that: a Lawvere–Tierney topology in $E$ is a morphism

j: \Omega \to \Omega

such that

$j true = true$ , equivalently $\id_\Omega \leq j: \Omega \to \Omega$ (‘if $p$ is true, then $p$ is locally true’)

$\array{ * &\stackrel{true}{\to}& \Omega \\ & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }$
$j j = j$ (? $p$ is locally locally true iff $p$ is locally true’);

$\array{ \Omega &\stackrel{j}{\to}& \Omega \\ & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }$
$j \circ \wedge = \wedge \circ (j \times j)$ (? $p \wedge q$ 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 partial order on $\Omega$ , and $\wedge: \Omega \times \Omega \to \Omega$ is the internal meet.

This appears for instance as (MacLaneMoerdijk, V 1.).

Example: cohesive topos

Revision on December 20, 2012 at 06:27:52 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn adjunctions, modalities and closures (Rev #1)

Contents

Adjunctions

Adjoint pairs in 22-categories

Mates

For quasicategories (In terms of cographs/correspondences)

1-categorical motivation

The Definition

Definition

Monads, Modalities and Closures

Example: Reflective subcategories

The Problem of Subtoposes

Theorem (Elephant A.4.3.9, p.192)

Lemma

Proof

Theorem

Relation to factorization systems, reflective subfibrations

Theorem

Module of a Monad

Modalities

Example: The Lawvere Tierney operator

Definition

Example: cohesive topos

Adjoint pairs in $2$ -categories