Computing Isotropy in Grothendieck...

Motivation and OverviewIntroducing Higher-Order Isotropy(Sequential) Colimits and Isotropy

Profunctors and Isotropy

Computing Isotropy in Grothendieck Toposes

Sakif Khan

University of Ottawa

[email protected]

August 12, 2016

Sakif Khan CT2016



Structure of Talk

1 Motivation and OverviewBrief Review of IsotropyFurther Motivation

2 Introducing Higher-Order Isotropy

3 (Sequential) Colimits and Isotropy

4 Profunctors and Isotropy

Sakif Khan CT2016



Brief Review of IsotropyFurther Motivation

Two Facets of Grothendieck Toposes

A (Grothendieck) topos E possesses simultaneously two kinds ofproperties: spatial and algebraic. We can deduce a few analogiesbetween these two aspects. Suppose X is an object in E .

Spatial Algebraic

Can take the class of subobjects of X Can take the slice E/XCollection of subobjects forms a poset Collection of automorphisms of E/X → E forms a groupGet a presheaf SubE(−) : Eop → Pos Get a presheaf Z(−) : Eop → Grp

SubE(−) represented by the subobject classifier Ω Z(−) represented by the isotropy group ZΩ is a locale internal to E Z is a group internal to E

Succinctly, the isotropy group of E encodes algebraic informationin much the same way that Ω encodes spatial information.

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Isotropy Group for Presheaf Toposes

Every Grothendieck topos has an isotropy group associated toit.

In particular, SetCop

contains such a group (for a smallcategory C) Z . In fact Z : Cop → Grp is the functor

Z (C ) = automorphisms of C/C → C

(see [FHS12]).

Explicitly, an element of Z (C ) is a family of automorphismsα : A→ AA∈C coherently lifting down to each other

A A

B B

α

α|f

f f

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Isotropy Group for Presheaf Toposes II

An easy example of this last point is given by a groupoid: Supposewe take objects A and B in a small groupoid G , an automorphismα : A→ A and a morphism f : B → A. Then we can always lift αalong f to an automorphism of B as in the diagram

A A

B B

α

f

f −1αf

f

by simple conjugation. For conjugation, it is clear that(α|f )|g = α|fg .

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Isotropy Group for Presheaf Toposes III

The maps in the image of Z (C )→ HomC(C ,C ) are calledisotropy maps on C.

We can take the collection of all isotropy maps I in C.

Induces an obvious quotient C C/I. This is the isotropyquotient of C, whose job is to trivialise the maps in I.

But can C/I itself have non-trivial isotropy maps? Yes!

How about iterated quotients C/In? Also yes. Leads to thenotion of higher-order isotropy (for small categories).

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Further Motivation

How can we actually tell when a (presheaf topos on a)category possesses higher-order isotropy?

Turning this question around, how might we build a categorywith a desired isotropy rank of some (ordinal) order?

Questions of isotropy rank are also questions about isotropygroups of categories.

Answering the first question answers: how do we computeisotropy groups of categories?

Answering the second question answers: how do we buildcategories with desired isotropy groups? – much the same wayEilenberg-MacLane spaces or Moore spaces are constructed tohave certain homotopy/homology groups.

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Higher-Order Isotropy

Let us make precise the notion of higher-order isotropy.

Definition

Note that for a small category C, we can keep taking isotropyquotients to get a sequence

C C/I C/I2 · · ·

which eventually stabilizes (for simple cardinality reasons) andwhere, for a limit ordinal µ, C/Iµ is the colimit lim←−

α<µ

C/Iα. We say

that C has λth-order isotropy if the chain stabilizes at stage λ.

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Isotropy Rank I

How does the isotropy of the category relate to automorphisms init?

Definition

Let C be a small category and ϕ : C → C an automorphism in C.The isotropy rank of ϕ, denoted ||ϕ||C, is defined by

||ϕ||C =

0 if ϕ = 1C∧λ | πλI(ϕ) = 1C if ∃λ > 0 such that πλI(ϕ) = 1C

−∞ otherwise.

Isotropy rank just says at which point in the isotropy chain ϕ getstrivialised.

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Isotropy Rank II

Lemma

The isotropy rank of the category C is the supremum

||C||I :=∨||ϕ||C | ϕ is an automorphism in C ≥ 0.

We also obtain a corresponding notion of preservation.

Definition

A functor F : C→ D preserves isotropy up to rank λ incase ||F (ϕ)||D = ||ϕ||C for all automorphisms ϕ ∈ Mor(C)with ||ϕ||C ≤ λ.

If we also have that ||C||I ≤ ||D||I and F preserves isotropyup to rank ||C||I , then F is said to simply preserve isotropyranks

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Isotropy Rank III

Put differently, a functor which preserves isotropy up to rank λ isone which can be lifted along isotropy quotient maps as indicatedin the diagram

C D

C/I D/I

......

C/Iλ D/Iλ

F

π1I π1

IF/I2

π2I π2

I

F/Iλ

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Building Models of Higher-Order Isotropy

Question

For a given ordinal λ, how can we build a category which hasλth-order isotropy?

Rough Answer

For finite-order isotropy, repeatedly take the collage of certainsimple profunctors. These fit together into a “nice” sequentialdiagram, the colimit of which gives ωth-order isotropy. Repeat forhigher successor and limit ordinals.

So, we need to develop some technology for manipulating/building(isotropy) automorphisms in small categories.

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Isotropy-preserving Functors

We will first look at how isotropy interacts with colimits. Butbefore that, a

Definition

A functor C→ D is isotropy-preserving if there is an inducedfunctor on isotropy quotients

C D

C/I D/I

where the vertical arrows are the canonical isotropy quotientfunctors. Moreover, it can be proven that there is at most onehorizontal filler making the square commute.

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Isotropy-preserving Functors II

The notion of isotropy-preservation is much coarser than thatof preserving isotropy up to specified isotropy rank.

Indeed, isotropy-preservation is just preserving isotropy up torank 1.

Let λ be the chain category of ordinals less than λ. Given asequential diagram F : λ→ Cat of categories and ordinalsα ≤ β < λ, we denote by F βα : F (α)→ F (β) the imageF (α ≤ β).

We say that the functor F βα : F (α)→ F (β) is a transition

map if F βα is full and injective on morphisms.

In particular, inclusions of categories are transition maps.

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Sequential colimits I

We are finally ready to state the

Theorem

Sequential colimits under diagrams with isotropy-preservingtransition maps commute with isotropy quotients.

and we obtain a

Corollary

Sequential colimits under diagrams with isotropy-preservinginclusions commute with isotropy quotients.

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Sequential colimits II

So, our corollary says that, under appropriate conditions, we have apicture

C0 C1 C2 · · · C

C0/I C1/I C2/I · · · C/I

where

all vertical arrows are isotropy quotient functors;

the inclusions in the bottom row are induced by theisotropy-preserving property of inclusions in the top row.

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Question

Can we propagate this diagram vertically so that higher quotientsof C are also computed as colimits of higher quotients?

Answer

In general, no. But recall that if isotropy ranks are preserved by afunctor, then we obtain a vertical tower of lifts for that functor.

If our inclusions have this additional property, then indeed,

C0 C1 C2 · · · C

C0/I C1/I C2/I · · · C/I

......

......

...

C0/Iλ C1/Iλ C2/Iλ · · · C/IλSakif Khan CT2016



Towards Profunctors

Our task of building models of higher-order isotropy has thus beenreduced to constructing a sequential diagram of categories withcertain nice properties – namely, each arrow in the sequence is aninclusion functor which preserves isotropy ranks. One way to buildsuch a sequence is to repeatedly take the collage of certain simplecategories. Let us recall a few basic facts.

Definition

A profunctor H : C 9 D is just a functor Dop × C→ Set. Itcorresponds to a two-sided codiscrete cofibration called the collage

C D

K

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Facts about Profunctors

A profunctor H is corepresentable ifH(d , c) = HomD(d ,F (c)) (and correspondingly on arrows) forsome functor F : C→ D.

The data of a codiscrete cofibration is precisely the data of afunctor K → ∆[1] such that the fibre over 0 is D and over 1 isC. Here, ∆[1] = 0→ 1.

We haveProf Cat/∆[1]

Profcorep Fib/∆[1]

'

'

where Fib indicates fibrations in the ordinary sense.

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Isotropy in Collages

We think of C as being “stacked on top” of D via the (corep)profunctor and the (hetero)morphisms from D to C in K as“connecting” objects in D to those in C.

Morphisms in C and D have isotropy ranks independently ofthe profunctor.

Forcing these morphisms to interact via a profunctor gives acategory – the collage – where isotropy ranks are dependenton both isotropy ranks in C and D.

How do we compute isotropy ranks in K given that we knowisotropy ranks in C and D? – Bit like computing fundamentalgroup of a space obtained by joining together two spaceswhose fundamental groups we already know.

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Isotropy in Collages II

The following result makes this intuition precise.

Theorem

Given an object c ∈ C, we have a pullback square

ZK (c) ZD(Fc)

ZC(c) AutC(c)

Ψc

Φc φc

ψc

in Grp.

The two projection maps are somewhat technical to define and weskip it for brevity.

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Lastly, we can relate isotropy quotients of C and D to that of K .

Proposition

The following are equivalent.

Each Φc in the pullback square

ZK (c) ZD(Fc)

ZC(c) X (c)

Ψc

Φc φc

ψc

is surjective.

The isotropy quotient C/I coincides with C/ ∼.

The isotropy quotient K/I is the collage of the functorF/I : C/I → D/I.

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Thank You

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References

Jean Benabou.

Distributors at work, June 2000.

Available athttp://www.mathematik.tu-darmstadt.de/ streicher/FIBR/DiWo.pdf.

Jonathan Funk, Pieter Hofstra, and Benjamin Steinberg.

Isotropy and crossed toposes.

Theory and Applications of Categories, 26(24):660–709, November 2012.

Available athttp://www.emis.de/journals/TAC/volumes/26/24/26-24abs.html.

Saunders MacLane and Ieke Moerdijk.

Sheaves in Geometry and Logic: A First Introduction to Topos Theory.

Universitext. Springer, October 1994.

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