The Groupoid Interpretation of Type Theory (1996)
| Venue: | In Venice Festschrift |
| Citations: | 16 - 1 self |
BibTeX
@INPROCEEDINGS{Hofmann96thegroupoid,
author = {Martin Hofmann and Thomas Streicher},
title = {The Groupoid Interpretation of Type Theory},
booktitle = {In Venice Festschrift},
year = {1996},
pages = {83--111},
publisher = {Oxford University Press}
}
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Abstract
ion and application Suppose that M 2 Tm(B). We define its abstraction A;B (M) 2 Tm(\Pi LF (A; B)) on objects by A;B (M)(fl)(a) = M(fl; a) A;B (M)(fl)(q) = M(id fl ; q) If p : fl ! fl 0 then we need a natural transformation A;B (M)(p) : p \Delta A;B (M)(fl) ! A;B (M)(fl 0 ) At object a 2 A(fl 0 ) it is given by M(p; id a ). Conversely, if M 2 Tm(\Pi(A; B)) we define a dependent object \Gamma1 A;B 2 Tm(B). Its object part is given by \Gamma1 A;B (M)(fl; a) = M(fl)(a) For the morphism part assume p : fl ! fl 0 and q : p \Delta a ! a 0 . We define \Gamma1 A;B (M)(p; q) = M(fl 0 )(q) ffi (id fl 0 ; q) \Delta M(p) p \Delta a We claim that \Gamma1 A;B (M)(p; q) : (p; q) \Delta \Gamma1 A;B (M)(fl; a) ! \Gamma1 A;B (M)(fl 0 ; a 0 ) as required. To see this, first note that M(fl 0 )(q) : (id fl 0 ; q) \Delta M 0 (fl 0 )(p \Delta a) !M 0 (fl 0 )(a 0 ) because q : p \Delta a ! a 0 . On the other hand M(p) : p \Delta M(fl) !M(fl 0 )...
