Bicategorical type theory: semantics and syntax

We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured bicategories. We start by developing the semantics, in the form of compr...

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Published inMathematical structures in computer science Vol. 33; no. 10; pp. 868 - 912
Main Authors Ahrens, Benedikt, North, Paige Randall, van der Weide, Niels
Format Journal Article
LanguageEnglish
Published 01.11.2023
Online AccessGet full text
ISSN0960-1295
1469-8072
1469-8072
DOI10.1017/S0960129523000312

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Abstract We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured bicategories. We start by developing the semantics, in the form of comprehension bicategories . Examples of comprehension bicategories are plentiful; we study both specific examples as well as classes of examples constructed from other data. From the notion of comprehension bicategory, we extract the syntax of bicategorical type theory, that is, judgment forms and structural inference rules. We prove soundness of the rules by giving an interpretation in any comprehension bicategory. The semantic aspects of our work are fully checked in the Coq proof assistant, based on the UniMath library.
AbstractList We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured bicategories. We start by developing the semantics, in the form of comprehension bicategories . Examples of comprehension bicategories are plentiful; we study both specific examples as well as classes of examples constructed from other data. From the notion of comprehension bicategory, we extract the syntax of bicategorical type theory, that is, judgment forms and structural inference rules. We prove soundness of the rules by giving an interpretation in any comprehension bicategory. The semantic aspects of our work are fully checked in the Coq proof assistant, based on the UniMath library.
Author Ahrens, Benedikt
North, Paige Randall
van der Weide, Niels
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10.1016/j.entcs.2019.09.012
10.1017/S0960129509007646
10.1007/BF00880041
10.1145/1960510.1960514
10.1007/BFb0066909
10.1016/S0022-4049(97)00129-1
10.1007/978-3-642-99902-4_2
10.1145/2103656.2103697
10.1145/3531130.3533363
10.1145/3531130.3533334
10.1016/j.jpaa.2013.11.002
10.1016/j.entcs.2011.09.026
10.1109/LICS.2017.8005124
10.1017/S0960129522000032
10.1007/BFb0074299
10.1007/978-3-319-15398-8
10.1109/LICS.2017.8005130
10.1109/LICS.2019.8785895
10.5281/zenodo.5846982
10.4171/JEMS/1050
10.1109/LICS.2019.8785708
10.1112/plms/pdq026
10.1109/LICS.1994.316071
10.21136/HS.2017.06
10.1017/S0960129514000486
10.1016/j.exmath.2019.02.004
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References S0960129523000312_ref18
S0960129523000312_ref17
S0960129523000312_ref39
S0960129523000312_ref16
S0960129523000312_ref38
S0960129523000312_ref15
S0960129523000312_ref37
S0960129523000312_ref19
Street (S0960129523000312_ref40) 1980; 21
Hirschowitz (S0960129523000312_ref22) 2013; 9
Lumsdaine (S0960129523000312_ref30) 2010; 6
S0960129523000312_ref32
S0960129523000312_ref10
S0960129523000312_ref31
S0960129523000312_ref14
S0960129523000312_ref36
S0960129523000312_ref35
S0960129523000312_ref13
S0960129523000312_ref12
S0960129523000312_ref34
S0960129523000312_ref33
S0960129523000312_ref11
S0960129523000312_ref29
S0960129523000312_ref4
S0960129523000312_ref28
S0960129523000312_ref7
S0960129523000312_ref27
S0960129523000312_ref26
S0960129523000312_ref6
S0960129523000312_ref9
Ahrens (S0960129523000312_ref3) 2019; 15
S0960129523000312_ref8
Bar (S0960129523000312_ref5) 2018; 14
S0960129523000312_ref1
S0960129523000312_ref2
Taylor (S0960129523000312_ref43) 1999; 59
S0960129523000312_ref21
S0960129523000312_ref20
S0960129523000312_ref42
S0960129523000312_ref41
S0960129523000312_ref25
S0960129523000312_ref46
S0960129523000312_ref24
S0960129523000312_ref45
S0960129523000312_ref23
S0960129523000312_ref44
References_xml – ident: S0960129523000312_ref46
  doi: 10.1145/3373718.3394794
– ident: S0960129523000312_ref31
  doi: 10.1016/j.entcs.2019.09.012
– ident: S0960129523000312_ref45
– ident: S0960129523000312_ref18
  doi: 10.1017/S0960129509007646
– ident: S0960129523000312_ref20
– ident: S0960129523000312_ref24
  doi: 10.1007/BF00880041
– ident: S0960129523000312_ref42
  doi: 10.1145/1960510.1960514
– ident: S0960129523000312_ref41
  doi: 10.1007/BFb0066909
– ident: S0960129523000312_ref21
  doi: 10.1016/S0022-4049(97)00129-1
– ident: S0960129523000312_ref8
– ident: S0960129523000312_ref19
  doi: 10.1007/978-3-642-99902-4_2
– ident: S0960129523000312_ref28
  doi: 10.1145/2103656.2103697
– ident: S0960129523000312_ref16
  doi: 10.1145/3531130.3533363
– ident: S0960129523000312_ref4
  doi: 10.1145/3531130.3533334
– ident: S0960129523000312_ref26
– ident: S0960129523000312_ref32
– ident: S0960129523000312_ref10
  doi: 10.1016/j.jpaa.2013.11.002
– ident: S0960129523000312_ref27
  doi: 10.1016/j.entcs.2011.09.026
– ident: S0960129523000312_ref36
– ident: S0960129523000312_ref38
– ident: S0960129523000312_ref15
  doi: 10.1109/LICS.2017.8005124
– ident: S0960129523000312_ref1
  doi: 10.1017/S0960129522000032
– volume: 21
  start-page: 111
  year: 1980
  ident: S0960129523000312_ref40
  article-title: Fibrations in bicategories
  publication-title: Cahiers de Topologie et Géométrie Différentielle Catégoriques
– ident: S0960129523000312_ref6
  doi: 10.1007/BFb0074299
– ident: S0960129523000312_ref14
  doi: 10.1007/978-3-319-15398-8
– ident: S0960129523000312_ref13
– ident: S0960129523000312_ref7
– volume: 14
  start-page: 1
  year: 2018
  ident: S0960129523000312_ref5
  article-title: Globular: an online proof assistant for higher-dimensional rewriting
  publication-title: Logical Methods in Computer Science
– ident: S0960129523000312_ref12
  doi: 10.1109/LICS.2017.8005130
– ident: S0960129523000312_ref9
– ident: S0960129523000312_ref33
  doi: 10.1109/LICS.2019.8785895
– volume: 15
  start-page: 1
  year: 2019
  ident: S0960129523000312_ref3
  article-title: Displayed categories
  publication-title: Logical Methods in Computer Science
– volume: 59
  volume-title: Cambridge Studies in Advanced Mathematics
  year: 1999
  ident: S0960129523000312_ref43
– ident: S0960129523000312_ref11
  doi: 10.5281/zenodo.5846982
– ident: S0960129523000312_ref25
  doi: 10.4171/JEMS/1050
– ident: S0960129523000312_ref17
  doi: 10.1109/LICS.2019.8785708
– volume: 6
  start-page: 1
  year: 2010
  ident: S0960129523000312_ref30
  article-title: Weak omega-categories from intensional type theory
  publication-title: Logical Methods in Computer Science
– volume: 9
  start-page: 1
  year: 2013
  ident: S0960129523000312_ref22
  article-title: Cartesian closed 2-categories and permutation equivalence in higher-order rewriting
  publication-title: Logical Methods in Computer Science
– ident: S0960129523000312_ref44
  doi: 10.1112/plms/pdq026
– ident: S0960129523000312_ref23
  doi: 10.1109/LICS.1994.316071
– ident: S0960129523000312_ref35
– ident: S0960129523000312_ref34
  doi: 10.21136/HS.2017.06
– ident: S0960129523000312_ref37
– ident: S0960129523000312_ref2
  doi: 10.1017/S0960129514000486
– ident: S0960129523000312_ref39
– ident: S0960129523000312_ref29
  doi: 10.1016/j.exmath.2019.02.004
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