Spahn adjunctions, modalities and closures (Rev #2, changes)

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Contents

Adjunctions

Adjoint pairs in 22-categories

Mates

For quasicategories (In terms of cographs/correspondences)

1-categorical motivation

Every profunctor?profunctor

k:C op×DS k : C^{op} \times D \to S

defines a category C* kDC *^k D with Obj(C* kD)=Obj(C)Obj(D)Obj(C *^k D) = Obj(C) \sqcup Obj(D) and

Hom C op×D(X,Y)={Hom C(X,Y) ifX,YC Hom D(X,Y) ifX,YD k(X,Y) ifXCandYD otherwise. 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?interval category

C* kDΔ 1. C *^k D \to \Delta^1 \,.

Now, every functor L:CDL : C \to D induces a profunctor?profunctor

k L(X,Y)=Hom D(L(X),Y) k_L(X,Y) = Hom_D(L(X), Y)

and every functor R:DCR : D \to C induces a profunctor?profunctor

k R(X,Y)=Hom C(X,R(Y)). k_R(X,Y) = Hom_C(X, R(Y)) \,.

The functors LL and RR are adjoint precisely if the profunctors?profunctors that they define in the above way are equal. This in turn is the case if C LD(D op R opC op) opC \star^L D \simeq (D^{op} \star^{R^{op}} C^{op})^{op}.

We say that C kDC \star^k D is the cograph of the functor?cograph of the functor kk. 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

(motivated from the discussion of correspondences in section 2.3.1)

to give a definition of adjunction between (infinity,1)-functors?(infinity,1)-functors.

Definition

Let CC and DD be quasi-categories?quasi-categories. An adjunction between CC and DD is

For more on this see

Monads, Modalities and Closures

Example: Reflective subcategories

The Problem of Subtoposes

Theorem (ElephantElephant? A.4.3.9, p.192)

Let (li):LiE(l\dashv i):L\stackrel{i}{\hookrightarrow}E be a reflective subcategory of a topos such that the monad ili\circ l is cartesiancartesian? (i.e. ili\circ l preserves pullbacks).

Then LL is a topos and ll preserves finite limits (i.e. (li)(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 (LR):ERH(L\dashv R):E\stackrel{\R}{\hookrightarrow} H be a reflective subcategory of a topos.

Then EE is a topos if LL preserves pullbacks in the image of a HR !a_H\circ R_! where

  • (a HY H):HPsh(H)(a_H\dashv Y_H):H\to Psh(H) is the left adjoint of the Yoneda embedding of HH.

  • R !:=Lan Y EY HRR_!:=Lan_{Y_E} Y_H\circ R is the left Kan extension of Y HRY_H\circ R along the Yoneda embedding of EE.

Psh(E) a EY E E R ! LR Psh(H) a HY H H\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
  1. The Yoneda embeddings of EE and HH both posess left adjoints: HH and EE are total: Since HH is a topos, HH is total, since EE is a reflective subcategory of a total category EE is total. By the adjoint functor theorem for total categories this implies that the Yoneda embeddings of EE and HH both posess left adjoints.

  2. We have a ELa HR !a_E\simeq L\circ a_H\circ R_!. If this composite is left exact it exhibits EE as a left exact localization of a category of presheaves and hence in this case EE is a topos.

  3. a HR !a_H\circ R_! sends colimits to limits, since R !R_! (as every Yoneda extension) commutes with colimits and a Ha_H as a left adjoint sends colimits to limits.

  4. Hence a ELa HR !a_E\simeq L\circ a_H\circ R_! is left exact iff LL preserves limits in the image of a HR !a_H\circ R_!.

  5. 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 LL preserves pullbacks in the image of a HR !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 subcategoryreflective subcategory? as Bashir Velebil). In particular EE is a topos.)

Relation to factorization systems, reflective subfibrations

Theorem

(1) The following statements are equivalent:

  • (E,M)(E,M) is a reflective factorization system in HH.

  • There is a reflective subcategory CHC\hookrightarrow H with reflector \sharp, EE is the class of morphisms whose \sharp-image is invertible in CC, and C=M/1C=M/1.

  • (E,M)(E,M) is a factorization system and EE satisfies 2-out -of-3.

  • (E,M)(E,M) is a factorization system and MM is the class of fibrant morphisms PAP\to A which as dependent types x:AP(x):Typex:A\dashv P(x): Type satisfy forallxinRsc(P(x))forall\, x \,in Rsc(P(x)).

  • For every HH-morphism f:ABf:A\to B satisfying: A\sharp A and B\sharp B are contractible, also for all bb we have hFiber(f,b)\sharp \, hFiber(f,b) is contractible.

isContr(A),isContr(B),f:AB,b:BisContr(hFiber(f,b))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)(E,M) is a factorization system in HH.

  • The class (E,M) ×:={M/x|xH,M/xH/xis.refl,refl.fact}(E,M)^\times:=\{M/x|x\in H,\M/x\hookrightarrow H/x\,is.refl,\,refl.fact\} is pullback-stable where refl.factrefl.fact means that each reflection is defined by (E,M)(E,M)-factorization.

  • (C.xH/x) xH(C.x\subseteq H/x)_{x\in H} is a pullback-stable system of reflective subcategories of slices of HH, and for every xx the class of objects of C.xC.x is closed under composition.

  • The class of types BB satisfying inRsc(B)in Rsc (B) is closed under dependent sums.

inRsc(A),forallx,inRsc(P(x))inRsc( x:AP(x))in Rsc(A), forall \, x,in Rsc(P(x))\vdash in Rsc (\sum_{x:A} P(x))

(3) The following statements are equivalent:

  • (C.xH/x) xH(C.x\subseteq H/x)_{x\in H} is a pullback-stable system of reflective subcategories of slices of HH, for every xx the class of objects of C.xC.x is closed under composition, and all reflectors commute with pullbacks.

  • The (by (2)) to (C.xH/x) xH(C.x\subseteq H/x)_{x\in H} corresponding factorization system (E,M)(E,M) is pullback stable.

Module of a Monad

Modalities

Example: The Lawvere Tierney operator

Definition

A Lawvere–Tierney topology in EE is (internallyinternally?) a closure operatorclosure operator? given by a left exactleft exact? idempotent monad on the internal meet-semilatticesemilattice? Ω\Omega.

This means that: a Lawvere–Tierney topology in EE is a morphismmorphism?

j:ΩΩ j: \Omega \to \Omega

such that

  1. jtrue=truej true = true, equivalently id Ωj:ΩΩ\id_\Omega \leq j: \Omega \to \Omega (‘if pp is true, then pp is locally true’)

    * true Ω true j Ω \array{ * &\stackrel{true}{\to}& \Omega \\ & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }
  2. jj=jj j = j (?pp is locally locally true iff pp is locally true’);

    Ω j Ω j j Ω \array{ \Omega &\stackrel{j}{\to}& \Omega \\ & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }
  3. j=(j×j)j \circ \wedge = \wedge \circ (j \times j) (?pqp \wedge q is locally true iff pp and qq are each locally true’)

    Ω×Ω Ω j×j j Ω×Ω Ω. \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 orderpartial order? on Ω\Omega, and :Ω×ΩΩ\wedge: \Omega \times \Omega \to \Omega is the internal meetmeet?.

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

Example: cohesive topos

Revision on December 20, 2012 at 16:01:09 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.