univalence sentence in Hindi

Examples

1. To formulate univalence as an axiom Voevodsky found a way to define " equivalences " syntactically that had the important property that the type representing the statement " f is an equivalence " was ( under the assumption of function extensionality ) (-1 )-truncated ( i . e . contractible if inhabited ).
2. In theory, a word like " univalence " could be coined as a synonym, but as Gandalf notes, " certainty " or " decisiveness " are already in common use . " Ambivalence " would be a synonym for " uncertainty " or " indecisiveness " . ?! carrots?! 12 : 56, 6 May 2011 ( UTC)
3. The participants also formulated a list of important open questions, such as whether the univalence axiom satisfies canonicity ( still open, although some special cases have been resolved positively ), whether the univalence axiom has nonstandard models ( since answered positively by Shulman ), and how to define ( semi ) simplicial types ( still open in MLTT, although it can be done in Voevodsky's Homotopy Type System ( HTS ), a type theory with two equality types ).
4. The participants also formulated a list of important open questions, such as whether the univalence axiom satisfies canonicity ( still open, although some special cases have been resolved positively ), whether the univalence axiom has nonstandard models ( since answered positively by Shulman ), and how to define ( semi ) simplicial types ( still open in MLTT, although it can be done in Voevodsky's Homotopy Type System ( HTS ), a type theory with two equality types ).
5. For instance, they noted that the groupoid model satisfies a rule they called " universe extensionality ", which is none other than the restriction to 1-types of the " univalence axiom " that Vladimir Voevodsky proposed ten years later . ( The axiom for 1-types is notably simpler to formulate, however, since a coherence notion of " equivalence " is not required . ) They also defined " categories with isomorphism as equality " and conjectured that in a model using higher-dimensional groupoids, for such categories one would have " equivalence is equality "; this was later proven by Ahrens, Kapulkin, and Shulman.