Computing Isotropy in Grothendieck Toposes Sakif Khan University of - - PowerPoint PPT Presentation

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Computing Isotropy in Grothendieck Toposes

Sakif Khan

University of Ottawa skhan172@uottawa.ca

August 12, 2016

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Structure of Talk

1

Motivation and Overview Brief Review of Isotropy Further Motivation

2

Introducing Higher-Order Isotropy

3

(Sequential) Colimits and Isotropy

4

Profunctors and Isotropy

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy Brief Review of Isotropy Further Motivation

Two Facets of Grothendieck Toposes

A (Grothendieck) topos E possesses simultaneously two kinds of properties: spatial and algebraic. We can deduce a few analogies between 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/X Collection of subobjects forms a poset Collection of automorphisms of E/X → E forms a group Get a presheaf SubE(−) : Eop → Pos Get a presheaf Z(−) : Eop → Grp SubE(−) represented by the subobject classifier Ω Z(−) represented by the isotropy group Z Ω is a locale internal to E Z is a group internal to E

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

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy Brief Review of Isotropy Further Motivation

Isotropy Group for Presheaf Toposes

Every Grothendieck topos has an isotropy group associated to it. In particular, SetCop contains such a group (for a small category 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 automorphisms {α : A → A}A∈C coherently lifting down to each other A A B B

α α|f f f

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy Brief Review of Isotropy Further Motivation

Isotropy Group for Presheaf Toposes II

An easy example of this last point is given by a groupoid: Suppose we take objects A and B in a small groupoid G, an automorphism α : A → A and a morphism f : B → A. Then we can always lift α along 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.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy Brief Review of Isotropy Further Motivation

Isotropy Group for Presheaf Toposes III

The maps in the image of Z(C) → HomC(C, C) are called isotropy 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 isotropy quotient 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 the notion of higher-order isotropy (for small categories).

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy Brief Review of Isotropy Further Motivation

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 category with a desired isotropy rank of some (ordinal) order? Questions of isotropy rank are also questions about isotropy groups of categories. Answering the first question answers: how do we compute isotropy groups of categories? Answering the second question answers: how do we build categories with desired isotropy groups? – much the same way Eilenberg-MacLane spaces or Moore spaces are constructed to have certain homotopy/homology groups.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

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 isotropy quotients to get a sequence C C/I C/I2 · · · which eventually stabilizes (for simple cardinality reasons) and where, 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.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Isotropy Rank I

How does the isotropy of the category relate to automorphisms in it? 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 =      if ϕ = 1C {λ | πλ

I(ϕ) = 1C}

if ∃λ > 0 such that πλ

I(ϕ) = 1C

−∞

• therwise.

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

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

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 in case ||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 isotropy up to rank ||C||I, then F is said to simply preserve isotropy ranks

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Isotropy Rank III

Put differently, a functor which preserves isotropy up to rank λ is

• ne which can be lifted along isotropy quotient maps as indicated

in the diagram C D C/I D/I . . . . . . C/Iλ D/Iλ

F π1

I

π1

I

F/I2 π2

I

π2

I

F/Iλ

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Building Models of Higher-Order Isotropy

Question For a given ordinal λ, how can we build a category which has λth-order isotropy? Rough Answer For finite-order isotropy, repeatedly take the collage of certain simple profunctors. These fit together into a “nice” sequential diagram, the colimit of which gives ωth-order isotropy. Repeat for higher successor and limit ordinals. So, we need to develop some technology for manipulating/building (isotropy) automorphisms in small categories.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Isotropy-preserving Functors

We will first look at how isotropy interacts with colimits. But before that, a Definition A functor C → D is isotropy-preserving if there is an induced functor on isotropy quotients C D C/I D/I where the vertical arrows are the canonical isotropy quotient

• functors. Moreover, it can be proven that there is at most one

horizontal filler making the square commute.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Isotropy-preserving Functors II

The notion of isotropy-preservation is much coarser than that

• f preserving isotropy up to specified isotropy rank.

Indeed, isotropy-preservation is just preserving isotropy up to rank 1. Let λ be the chain category of ordinals less than λ. Given a sequential diagram F : λ → Cat of categories and ordinals α ≤ β < λ, we denote by F β

α : F(α) → F(β) the image

F(α ≤ β). We say that the functor F β

α : F(α) → F(β) is a transition

map if F β

α is full and injective on morphisms.

In particular, inclusions of categories are transition maps.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Sequential colimits I

We are finally ready to state the Theorem Sequential colimits under diagrams with isotropy-preserving transition maps commute with isotropy quotients. and we obtain a Corollary Sequential colimits under diagrams with isotropy-preserving inclusions commute with isotropy quotients.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Sequential colimits II

So, our corollary says that, under appropriate conditions, we have a picture 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 the isotropy-preserving property of inclusions in the top row.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Question Can we propagate this diagram vertically so that higher quotients

• f C are also computed as colimits of higher quotients?

Answer In general, no. But recall that if isotropy ranks are preserved by a functor, 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/Iλ

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Towards Profunctors

Our task of building models of higher-order isotropy has thus been reduced to constructing a sequential diagram of categories with certain nice properties – namely, each arrow in the sequence is an inclusion functor which preserves isotropy ranks. One way to build such a sequence is to repeatedly take the collage of certain simple

• categories. Let us recall a few basic facts.

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

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Facts about Profunctors

A profunctor H is corepresentable if H(d, c) = HomD(d, F(c)) (and correspondingly on arrows) for some functor F : C → D. The data of a codiscrete cofibration is precisely the data of a functor K → ∆[1] such that the fibre over 0 is D and over 1 is

• C. Here, ∆[1] = 0 → 1.

We have Prof Cat/∆[1] Profcorep Fib/∆[1]

≃ ≃

where Fib indicates fibrations in the ordinary sense.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

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 of the profunctor. Forcing these morphisms to interact via a profunctor gives a category – the collage – where isotropy ranks are dependent

• n both isotropy ranks in C and D.

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

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

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 we skip it for brevity.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

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 functor F/I : C/I → D/I.

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

Thank You

Sakif Khan CT2016

Motivation and Overview Introducing Higher-Order Isotropy (Sequential) Colimits and Isotropy Profunctors and Isotropy

References

Jean B´ enabou. Distributors at work, June 2000. Available at http://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 at http://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.

Sakif Khan CT2016