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 Motivation and Overview 1 Brief Review of Isotropy Further Motivation Introducing Higher-Order Isotropy 2 (Sequential) Colimits and Isotropy 3 Profunctors and Isotropy 4 Sakif Khan CT2016
Motivation and Overview Introducing Higher-Order Isotropy Brief Review of Isotropy (Sequential) Colimits and Isotropy Further Motivation Profunctors and Isotropy 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 Sub E ( − ) : E op → Pos Get a presheaf Z ( − ) : E op → Grp Sub E ( − ) 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 Brief Review of Isotropy (Sequential) Colimits and Isotropy Further Motivation Profunctors and Isotropy Isotropy Group for Presheaf Toposes Every Grothendieck topos has an isotropy group associated to it. In particular, Set C op contains such a group (for a small category C ) Z . In fact Z : C op → 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 α A A f f B B α | f Sakif Khan CT2016
Motivation and Overview Introducing Higher-Order Isotropy Brief Review of Isotropy (Sequential) Colimits and Isotropy Further Motivation Profunctors and Isotropy 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 α A A f f B B f − 1 α 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 Brief Review of Isotropy (Sequential) Colimits and Isotropy Further Motivation Profunctors and Isotropy Isotropy Group for Presheaf Toposes III The maps in the image of Z ( C ) → Hom C ( 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 / I n ? Also yes. Leads to the notion of higher-order isotropy (for small categories). Sakif Khan CT2016
Motivation and Overview Introducing Higher-Order Isotropy Brief Review of Isotropy (Sequential) Colimits and Isotropy Further Motivation Profunctors and Isotropy 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 / I 2 C C / I · · · which eventually stabilizes (for simple cardinality reasons) and where, for a limit ordinal µ , C / I µ is the colimit lim C / I α . We say ← − α<µ that C has λ th -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 0 if ϕ = 1 C � { λ | π λ if ∃ λ > 0 such that π λ || ϕ || C = I ( ϕ ) = 1 C } I ( ϕ ) = 1 C −∞ otherwise. 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 one which can be lifted along isotropy quotient maps as indicated in the diagram F C D π 1 π 1 I I F / I 2 C / I D / I π 2 π 2 I I . . . . . . F / I λ C / I λ D / I λ 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 λ th -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 ω th -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 of 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 C 0 C 1 C 2 · · · C C 0 / I C 1 / I C 2 / 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
Recommend
More recommend
