Constructing Categories of Corelations Brendan Fong (MIT) - - PowerPoint PPT Presentation

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Constructing Categories of Corelations Brendan Fong (MIT) Octoberfest Carnegie Mellon University 28 October 2017 I. Motivation This is a presentation of the category LinRel k ( x ) . How do we prove this? How do we prove this? Consider the

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Constructing Categories of Corelations

Brendan Fong (MIT)

Octoberfest Carnegie Mellon University 28 October 2017

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  • I. Motivation
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This is a presentation of the category LinRelk(x).

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How do we prove this?

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How do we prove this? Consider the following square. Vect + Vect

  • p

Span(Vect) Cospan(Vect) LinRel

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How do we prove this? Consider the following square. Vect + Vect

  • p

Span(Vect) Cospan(Vect) LinRel This is a pushout square in the category of props.

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How do we prove this? Consider the following square. Vect + Vect

  • p

Span(Vect) Cospan(Vect) LinRel This is a pushout square in the category of props. Linear relations interpret diagrams of linear maps ←

  • →←
  • →←
  • where we may compose by function composition, pullback, and

pushout.

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How do we prove this? Consider the following square. Vect + Vect

  • p

Span(Vect) Cospan(Vect) LinRel This is a pushout square in the category of props. Linear relations interpret diagrams of linear maps ←

  • →←
  • →←
  • where we may compose by function composition, pullback, and

pushout. This leads to a presentation of LinRel.

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Colimits combine systems.

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Colimits combine systems. Monoidal categories of cospans allow construction of all finite colimits, via

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Colimits combine systems. Monoidal categories of cospans allow construction of all finite colimits, via

  • composition (pushout),
  • monoidal product (binary coproducts), and
  • monoidal unit (initial object).
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Colimits combine systems. Monoidal categories of cospans allow construction of all finite colimits, via

  • composition (pushout),
  • monoidal product (binary coproducts), and
  • monoidal unit (initial object).

Thus cospan categories provide useful language for system interconnection.

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Colimits combine systems. Monoidal categories of cospans allow construction of all finite colimits, via

  • composition (pushout),
  • monoidal product (binary coproducts), and
  • monoidal unit (initial object).

Thus cospan categories provide useful language for system interconnection. However, combining systems using colimits indiscriminately accumulates information.

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Consider cospans in FinSet. X N Y

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Consider cospans in FinSet. X N Y

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Consider cospans in FinSet. X N Y If we think about these as circuits, all we care about is the induced equivalence relation on X + Y .

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X N Z M Y

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X N Z M Y = X Z

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X N Z M Y = X Z Cospans accumulate internal structure (witnesses for ‘empty equivalence classes’).

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X N Z M Y = X Z Cospans accumulate internal structure (witnesses for ‘empty equivalence classes’). Corelations forget this.

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Factorisation hides internal structure.

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Factorisation hides internal structure. A factorisation system (E,M) comprises subcategories E, M such that

  • E and M contain all isomorphisms
  • every f admits factorisation f = m ○ e.
  • we have the universal property:

e

  • u
  • m
  • ∃!s
  • v
  • e′
  • m′
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Factorisation hides internal structure. A factorisation system (E,M) comprises subcategories E, M such that

  • E and M contain all isomorphisms
  • every f admits factorisation f = m ○ e.
  • we have the universal property:

e

  • u
  • m
  • ∃!s
  • v
  • e′
  • m′

For example, epi–mono factorisation systems (like in FinSet).

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A corelation is an equivalence class of cospans, where two cospans are equivalent if N N ′ X Y.

m

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A corelation is an equivalence class of cospans, where two cospans are equivalent if N N ′ X Y.

m

We may represent each corelation by a cospan such that X + Y → N lies in E.

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A corelation is an equivalence class of cospans, where two cospans are equivalent if N N ′ X Y.

m

We may represent each corelation by a cospan such that X + Y → N lies in E. When M is stable under pushout, composition by pushout defines a category Corel(C).

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What is the link?

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What is the link? Vect + Vect

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Span(Vect) Cospan(Vect) LinRel ≅ Corel(Vect)

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What is the link? Vect + Vect

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Span(Vect) Cospan(Vect) LinRel ≅ Corel(Vect) So we claim:

  • I. Corelations model system interconnection and
  • II. A universal property is useful for computing presentations.
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What is the link? Vect + Vect

  • p

Span(Vect) Cospan(Vect) LinRel ≅ Corel(Vect) So we claim:

  • I. Corelations model system interconnection and
  • II. A universal property is useful for computing presentations.

Does this universal construction generalise to other corelation categories?

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  • II. A Universal Property for

Corelations

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Cospan(C) Corel(C)

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? + ?

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Span(?) Cospan(C) Corel(C)

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? + ?

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Span(?) Cospan(C) Corel(C) A functor Span(C) → Corel(C) does not in general exist. Under what conditions might it exist?

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Define a map SpanC(A) → Corel(C) by taking pushouts.

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

m?

These two cospans represent the same corelation when the canonical map lies in M.

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

m?

These two cospans represent the same corelation when the canonical map lies in M. Call this the pullback–pushout property (with respect to M).

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Define a map SpanC(A) → Corel(C) by taking pushouts. When is this functorial?

m?

These two cospans represent the same corelation when the canonical map lies in M. Call this the pullback–pushout property (with respect to M). When A obeys the pullback–pushout property, then there exists a functor SpanC(A) → Corel(C).

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Theorem

Suppose a category C has

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Theorem

Suppose a category C has

  • pushouts and pullbacks
  • a factorisation system (E,M) with M ⊆ monos, stable un-

der pushout

  • such that M obeys the pullback–pushout property.
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Theorem

Suppose a category C has

  • pushouts and pullbacks
  • a factorisation system (E,M) with M ⊆ monos, stable un-

der pushout

  • such that M obeys the pullback–pushout property.

Then we have a pushout square in Cat: M +∣M∣ Mop SpanC(M) Cospan(C) Corel(C)

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Theorem: generalising M

Suppose C has

  • pushouts and pullbacks
  • a factorisation system with M ⊆ monos, stable under

pushout

  • a subcategory A ⊇ M, stable under pullback, obeying the

pullback–pushout property. Then we have a pushout square in Cat: A +∣A∣ A

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SpanC(A) Cospan(C) Corel(C)

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Corollary: abelian case

Let C be an abelian category. This has a (co)stable epi–mono factorisation system. The theorem can also be extended to monoidal categories, by requiring that the monoidal product preserve pushouts in C and pullbacks in A, and that M and A are closed under the monoidal product.

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Corollary: abelian case

Let C be an abelian category. This has a (co)stable epi–mono factorisation system. We have a pushout square in Cat: C +∣C∣ C

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Span(C) Cospan(C) Corel(C) ≅ Rel(C)

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Corollary: abelian case

Let C be an abelian category. This has a (co)stable epi–mono factorisation system. We have a pushout square in Cat: C +∣C∣ C

  • p

Span(C) Cospan(C) Corel(C) ≅ Rel(C) The theorem can also be extended to monoidal categories, by requiring that the monoidal product preserve pushouts in C and pullbacks in A, and that M and A are closed under the monoidal product.

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Examples

Corelations (Equivalence relations): Inj +● Inj

  • p

Span(Inj) Cospan(FinSet) Corel(FinSet) Partial equivalence relations: PInj +● PInj

  • p

Span(PInj) Cospan(ParFunc) PER ≅ Corel(ParFunc)

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Examples

Linear relations: Vect +● Vect

  • p

Span(Vect) Cospan(Vect) Corel(Vect) ≅ LinRel Discrete time, linear, time-invariant, dynamical systems over k: SpltM +● SpltM

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Span(SpltM) Cospan(Matk[s,s−1]) Corel(Matk[s,s−1])

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Examples

Let T be a comonad on Set such that T and T 2 both preserve pullbacks of regular monos. Then the category SetT of coal- gebras over T obeys the theorem with respect to (epis, regular monos).

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Examples

Let T be a comonad on Set such that T and T 2 both preserve pullbacks of regular monos. Then the category SetT of coal- gebras over T obeys the theorem with respect to (epis, regular monos). This property is obeyed by the cofree comonad on the double finite power set functor, which has been used to model logic programs.

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Theorem: dual case

Suppose a category C has

  • pushouts and pullbacks
  • a factorisation system (E,M) with E ⊆ epis, stable under

pullback

  • such that E obeys the pullback–pushout property.

Then we have a pushout square in Cat: E +∣E∣ E op Span(C) Cospan(E) Rel(C)

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Non-example: Relations

Surj does not obey pushout–pullback property. Surj + Surj

  • p

Span(FinSet) Cospan(Surj) Rel(FinSet)

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Non-example: Relations

Surj does not obey pushout–pullback property. Surj + Surj

  • p

Span(FinSet) Cospan(Surj) Rel(FinSet) 3 2 2

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Non-example: Relations

Surj does not obey pushout–pullback property. Surj + Surj

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Span(FinSet) Cospan(Surj) Rel(FinSet) 3 2 2 1

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Non-example: Relations

Surj does not obey pushout–pullback property. Surj + Surj

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Span(FinSet) Cospan(Surj) Rel(FinSet) 3 4 2 2 1

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Non-example: Relations

Surj does not obey pushout–pullback property. Surj + Surj

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Span(FinSet) Cospan(Surj) Rel(FinSet) 3 4 2 2 1 Not an epi!

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Non-example: Relations

Surj does not obey pushout–pullback property. Surj + Surj

  • p

Span(FinSet) Cospan(Surj) Rel(FinSet) 3 4 2 2 1 Not an epi! (We cannot construct Rel = Rel(FinSet) as a pushout.)

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To recap:

  • I. Corelations model system interconnection
  • II. Categories of corelations can be constructed as a pushout of span

and cospan categories.

  • III. This helps derive presentations.
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I thank Fabio Zanasi for collaborating on this work. Thank you for listening. For more: http://www.brendanfong.com/