Categorical semantics

From LLWiki
(Difference between revisions)
Jump to: navigation, search
Line 1: Line 1:
TODO: why categories? how to extract categorical models? etc.
+
Constructing denotational models of linear can be a tedious work. Categorical model are useful to identify the fundamental structure of these models, and thus simplify and make more abstract the elaboration of those models.
  +
  +
TODO: why categories? how to extract categorical models? etc.
   
 
See <ref>{{BibEntry|bibtype=book|author=MacLane, Saunders|title=Categories for the Working Mathematician|publisher=Springer Verlag,year=1971,volume=5,series=Graduate Texts in Mathematics}}</ref>for a more detailed introduction to category theory.
 
See <ref>{{BibEntry|bibtype=book|author=MacLane, Saunders|title=Categories for the Working Mathematician|publisher=Springer Verlag,year=1971,volume=5,series=Graduate Texts in Mathematics}}</ref>for a more detailed introduction to category theory.
Line 8: Line 8:
   
 
{{Definition|title=Monoidal category|
 
{{Definition|title=Monoidal category|
A ''monoidal category'' <math>(\mathcal{C},\otimes,I)</math> is a category <math>\mathcal{C}</math> equipped with
+
A ''monoidal category'' <math>(\mathcal{C},\otimes,I,\alpha,\lambda,\rho)</math> is a category <math>\mathcal{C}</math> equipped with
 
* a functor <math>\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}</math> called ''tensor product'',
 
* a functor <math>\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}</math> called ''tensor product'',
 
* an object <math>I</math> called ''unit object'',
 
* an object <math>I</math> called ''unit object'',
* three natural isomorphisms of components
+
* three natural isomorphisms <math>\alpha</math>, <math>\lambda</math> and <math>\rho</math>, called respectively ''associator'', ''left unitor'' and ''right unitor'', whose components are
 
: <math>
 
: <math>
 
\alpha_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)
 
\alpha_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)
Line 19: Line 19:
 
\rho_A:A\otimes I\to A
 
\rho_A:A\otimes I\to A
 
</math>
 
</math>
called respectively ''associator'', ''left unitor'' and ''right unitor'',
 
 
 
such that
 
such that
 
* for every objects <math>A,B,C,D</math> in <math>\mathcal{C}</math>, the diagram
 
* for every objects <math>A,B,C,D</math> in <math>\mathcal{C}</math>, the diagram
Line 38: Line 36:
 
:<math>\gamma_{B,A}\circ\gamma_{A,B}=A\otimes B</math>
 
:<math>\gamma_{B,A}\circ\gamma_{A,B}=A\otimes B</math>
 
for every objects <math>A</math> and <math>B</math>.
 
for every objects <math>A</math> and <math>B</math>.
  +
}}
  +
  +
== Modeling the additives ==
  +
{{Definition|title=Product|
  +
}}
  +
  +
{{Definition|title=Monoid|
  +
}}
  +
  +
{{Property|
  +
Categories with products vs monoidal categories.
 
}}
 
}}
   

Revision as of 18:40, 23 March 2009

Constructing denotational models of linear can be a tedious work. Categorical model are useful to identify the fundamental structure of these models, and thus simplify and make more abstract the elaboration of those models.

TODO: why categories? how to extract categorical models? etc.

See [1]for a more detailed introduction to category theory.

Contents

Modeling IMLL

A model of IMLL is a closed symmetric monoidal category. We recall the definition of these categories below.

Definition (Monoidal category)

A monoidal category (\mathcal{C},\otimes,I,\alpha,\lambda,\rho) is a category \mathcal{C} equipped with

  • a functor \otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C} called tensor product,
  • an object I called unit object,
  • three natural isomorphisms α, λ and ρ, called respectively associator, left unitor and right unitor, whose components are

\alpha_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)
\qquad
\lambda_A:I\otimes A\to A
\qquad
\rho_A:A\otimes I\to A

such that

  • for every objects A,B,C,D in \mathcal{C}, the diagram

commutes,

  • for every objects A and B in \mathcal{C}, the diagrams

commute.

Definition (Braided, symmetric monoidal category)

A braided monoidal category is a category together with a natural isomorphism of components

\gamma_{A,B}:A\otimes B\to B\otimes A

called braiding, such that the two diagrams

UNIQ7eccaa7c5e5ea4da-math-00000011-QINU

commute for every objects A, B and C.

A symmetric monoidal category is a braided monoidal category in which the braiding satisfies

\gamma_{B,A}\circ\gamma_{A,B}=A\otimes B

for every objects A and B.

Modeling the additives

Definition (Product)


Definition (Monoid)


Property

Categories with products vs monoidal categories.

Modeling IMALL

Modeling negation

Definition (*-autonomous category)

TODO

References

  1. MacLane, Saunders. Categories for the Working Mathematician.
Personal tools