Itegories Itegories J.R.B. Cockett Department of Computer Science University of Calgary Alberta, Canada robin@cpsc.ucalgary.ca (work with: Pieter Hofstra, Chad Nester) Aviero, June 2015
Itegories Introduction Restriction category Basics Extensivity Independence Disjoins Constructions Itegories Basics Traces Cartesian itegories Computability Stack objects Abstract machine
Itegories Introduction WHAT IS THIS TALK ABOUT? Answer: a very elementary setting for describing sequential computation ... The semantics of flow diagrams ... (an old topic) (Dana Scott, Elgot, Steve Bloom, Ernie Manes, Bob Walters, ...)
Itegories Introduction To the memory of Bob Walters: He would not have described it this way nor necessarily would he have approved but an itegory is something he certainly thought about ...
Itegories Introduction What is an Itegory An itegory is a disjoin restriction category .... with a Kleene wand. Idea: (a) Disjoint unions of (partial) maps: f ⊔ g (b) A Kleene wand f : A − → A g : A − → B f ⊥ g f ∗ | g : A − → B f ⊥ g means f is disjoint from g ... Kleene wand is a way of specifying a particle style trace. Begin with restriction categories ...
Itegories Restriction category Basics A restriction category is a category X equipped with a restriction : f A − − → B A − − → A f [R.1] f f = f [R.2] f g = g f [R.3] f g = f g [R.4] f g = fgf A map f is said to be total when f = 1. Lemma (i) f f = f (iv) f g = f g (ii) f is monic then f = 1 (v) f = f (iii) fg = f g (vi) f , g total implies fg total.
Itegories Restriction category Basics COMPLETENESS: Every restriction category is a full subcategory of a real category of partial maps! (A convenient algebraic framework for partial maps ...)
� � � Itegories Restriction category Basics Partial products A restriction category has a partial terminal object , 1, in case for each object A there is a total map ! A : A − → 1 such that given any other map h : A − → 1 we have h = h ! A . A restriction category has partial products in case for each pair of object A and B there is an object A × B and total maps π 0 : A × B − → A and π 1 : A × B − → B such that given any pair of maps f : X − → A and g : X − → B there is a map � f , g � : X − → A × B with � f , g � π 0 = gf and � f , g � π 1 = f g . A � ❋ ❋ ⑧ ❋ ⑧ ❋ π 0 f ⑧ ❋ ⑧ ≥ ❋ ⑧ ❋ ⑧ ❋ ⑧ ❋ ⑧ � f , g � X A × B ❄ ❄ � ①①①①①①①①① ❄ ❄ ❄ ≥ ❄ g π 1 ❄ ❄ B
Itegories Restriction category Basics Cartesian restriction categories A restriction category which has partial products is a cartesian restriction category . (a.k.a. “P-category” developed by Robinson and Rossolini) Any Cartesian restriction category has a total category with real products and terminal object.
Itegories Restriction category Basics Examples of restriction categories 1. The category of sets and partial maps Par(Set , monic ). 2. Any partial map category: a favourite is CRing op with localizations L . Then Par(CRing op , L ) is a restriction category (of use in algebraic geometry). 3. The category of topological spaces with partial maps defined on an open subset and a continuous map on that subset. 4. The category sSLat op with stable maps (i.e. binary meet preserving maps). The category of locales with stable maps (i.e. binary meet and join preserving maps). 5. Partial recursive maps on the natural numbers. 6. Given any partial algebraic theory T there is a classifying cartesian restriction category C ( T ) with a generic model of the partial algebraic theory. For example, there is a generic partial combinatory algebra which lives in its own environment and gives a generic version of computability.
� � � � Itegories Restriction category Basics Partial Combinator Algebras (PCAs) With partial product one can express arbitrary partial algebras. Here is a Partial Combinatory Algebra (PCA): • : A × A − → A k , s : 1 − → A such that k = 1 1 and s = 1 1 k × 1 × 1 s × 1 × 1 × 1 � A × A × A � A × A × A × A A × A ( A × A ) × A ▼ ▼ ▼ ▼ ▼ ▼ • 2 ▼ θ × • 3 ▼ π 0 ▼ ▼ ▼ ▼ A � A ( A × A ) × ( A × A ) ( •×• ) • (s × 1 × 1) • 2 = 1 Remark: Given an (internal) PCA one can not only express all computations but also one has the usual undecidability results associated with computability.
Itegories Restriction category Extensivity Restriction zeros A restriction category has a restriction zeros if for each pair of objects A and B there is a map 0 : A − → B such that ◮ f 0 g = 0 ◮ 0 = 0 If a zero map splits then the category has a zero object. If a restriction category has restriction zeros then it is enriched over pointed sets. Note that this category is also the category of sets and partial maps. Sets and partial maps have restriction zeros.
Itegories Restriction category Extensivity Coproducts and extensivity An extensive restriction category has coproducts, restriction zeros, and a decision operator : f : X − → Y + Z � f � : X − → X + X [Dec.1] ∇� f � = f [Dec.2] ( f + f ) � f � = ( σ 0 + σ 1 ) f d : X − → X + X is a decision in case d = � d � . Binary decisions imply n -ary decisions d : A − → A + ... + A (these satisfy ∇ d = d and ( d + ... + d ) d = ( σ 1 + ... + σ n ) d ). Extensivity implies unique decomposition as the coproduct is a pre-biproduct: (( σ ( − 1) σ 0 ) + ( σ ( − 1) σ 1 ) f = ( σ ( − 1) + σ ( − 1) f = )( f + f ) � f � 0 1 0 1 ∇ ( σ ( − 1) f + σ ( − 1) f ) � f � = ( σ ( − 1) f ⊔ σ ( − 1) = f ) 0 1 0 1
Itegories Restriction category Extensivity Extensivity cont. Why is extensivity important? Every map between coproducts an be written as a matrix .. X 1 + ... + X n − − − − − − − − − − − − − − − − → Y 1 + ... + Y m a 11 ... a 1 n A = ... a m 1 ... a mn where the i th rows must be separated by a decision d i such that a ij = σ j d i
Itegories Restriction category Independence Independence An independence space ( X , X ) is a set with a set of finite subsets X ⊆ P f ( X ) which are down-closed and contain all singletons. A morphism f : ( X , X ) − → ( Y , Y ) between independence spaces is a partial map f : X − → Y which, when restricted to any independent subset, becomes a partial isomorphism. Each X ′ ∈ X determines a restriction idempotent e X ′ : X − → X , we require that e X ′ f is a partial isomorphism and ( e X ′ f ) ( − 1) determines an independent subset of Y . independent subsets cannot be “squashed” by maps ... The category of independence spaces is an extensive restriction category. It does not have partial products but it has various tensors ....
� � Itegories Restriction category Independence Independence spaces Have the following (non-symmetric) tensor product: ( A , A ) ⊳ ( B , B ) = ( A × B , A ⊳ B ) where X ∈ A ⊳ B if and only if { a |∃ b . ( a , b ) ∈ X } ∈ A & X a = { b | ( a , b ) ∈ X } ∈ B b 11 b m 1 . . . . . . b 1 n · · · b mn m a 1 · · · a m Every object has a naturally coassociative comultiplication ...
Itegories Restriction category Independence Enriching in Independence Require that each X ( A , B ) ∗ (remove zero) is an independence space and ( ) : X ( A , B ) ∗ − → X ( A , A ) ∗ and composition are independence space maps. ◮ Parallel arrows { f 1 , ..., f n } are independent if and only if { f 1 , ..., f n } is. ◮ If { f 1 , ..., f n } are independent and f i ≤ f ′ i then { f ′ 1 , ..., f ′ n } are independent. ◮ f and g are independent then f g = 0. Proof (of last) If { f , g } is independent then { ( f , f ) , ( f , g ) , ( g , f ) , ( g , g ) } is independent in X ( A , A ) ⊳ X ( A , A ) but under composition f g = g f so these non-diagonal pairs are squashed under composition so cannot be defined. This means that these composites must be zero.
Itegories Restriction category Independence Examples of independences ◮ Given any restriction category X with a restriction zero declare the independent sets on X ( A , B ) ∗ (where we remove the zero) to be the singleton sets. ◮ Given any restriction category X with a restriction zero declare the independent sets on X ( A , B ) ∗ to consist of the finite sets X ′ ⊆ X ( A , B ) ∗ such that for each distinct pair f , g ∈ X ′ we have f g = 0. ◮ (Separation) Given any extensive category X declare { f 1 , ..., f n } ⊆ X ( A , B ) ∗ independent in case there is a decision → A + ... + A such that f i ≤ σ ( − 1) d : A − d . i Does separation in extensive categories cover all examples? Answer: Yes
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