Page 638 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 638
LOGIC AND MATHEMATICAL ASPECTS OF COMPUTER SCIENCE
Aggregation of individual rankings through fusion functions: criticism
and optimality analysis
María Jesús Campión, Mjesus.campion@unavarra.es
Universidad Pública de Navarra, Spain
Coauthors: Humberto Bustince, Benjamín Bedregal, Ivanosca Andrade da Silva,
Javier Fernández, Esteban Induráin, Armajac Raventos Pujol, Regivan Santiago
Our main idea is to analyze from a theoretical and normative point of view different methods to
aggregate individual rankings. To do so, first we introduce the concept of a general mean on an
abstract set. This new concept conciliates the Social Choice where well-known impossibility
results as the Arrovian ones are encountered and the Decision-Making approaches where the
necessity of fusing rankings is unavoidable. Moreover it gives rise to a reasonable definition of
the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean.
Then we will introduce some methods to build ranking fusion functions, paying a special atten-
tion to the use of score functions, and pointing out the equivalence between ranking and scoring.
To conclude, we prove that any ranking fusion function introduces a partial order on rankings
implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and
different methods of aggregation, so that in practice one should look for the maximal elements
with respect to such orders defined on rankings IEEE.
Towards Non-Presentable Models of Homotopy Type Theory
Nima Rasekh, nima.rasekh@epfl.ch
EPFL, Switzerland
One important aspect of homotopy type theory is the construction of models: (∞, 1)-categories
in which we can interpret the axioms of our type theory. Studying various models can help us
discern which statements can and cannot be proven with our given axiomatization.
Due to a result by Shulman, we already know that every Grothendieck (∞, 1)-topos is a
model for homotopy theory. However, we do not expect all models to be a Grothendieck (∞, 1)-
topos and in particular, we anticipate non-presentable models of homotopy type theory.
The goal of this work is to take a first step towards showing the existence of non-presentable
models of homotopy type theory, by constructing a non-presentable elementary (∞, 1)-topos
. Elementary (∞, 1)-toposes share many features with Grothendieck (∞, 1)-toposes (such as
descent, universes, natural number objects, ... ), but are not required to be presentable and thus
can include examples that are not Grothendieck (∞, 1)-toposes.
We will construct such examples via the filter construction. Generalizing a result from 1-
category theory, we prove that for every elementary (∞, 1)-topos E and filter of subobjects
Φ, we can construct an elementary (∞, 1)-topos Φ E, which is in fact not presentable if the
filter Φ is not principal. We will apply this result to the (∞, 1)-category of Kan complexes to
construct non-presentable examples of elementary (∞, 1)-toposes.
636
Aggregation of individual rankings through fusion functions: criticism
and optimality analysis
María Jesús Campión, Mjesus.campion@unavarra.es
Universidad Pública de Navarra, Spain
Coauthors: Humberto Bustince, Benjamín Bedregal, Ivanosca Andrade da Silva,
Javier Fernández, Esteban Induráin, Armajac Raventos Pujol, Regivan Santiago
Our main idea is to analyze from a theoretical and normative point of view different methods to
aggregate individual rankings. To do so, first we introduce the concept of a general mean on an
abstract set. This new concept conciliates the Social Choice where well-known impossibility
results as the Arrovian ones are encountered and the Decision-Making approaches where the
necessity of fusing rankings is unavoidable. Moreover it gives rise to a reasonable definition of
the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean.
Then we will introduce some methods to build ranking fusion functions, paying a special atten-
tion to the use of score functions, and pointing out the equivalence between ranking and scoring.
To conclude, we prove that any ranking fusion function introduces a partial order on rankings
implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and
different methods of aggregation, so that in practice one should look for the maximal elements
with respect to such orders defined on rankings IEEE.
Towards Non-Presentable Models of Homotopy Type Theory
Nima Rasekh, nima.rasekh@epfl.ch
EPFL, Switzerland
One important aspect of homotopy type theory is the construction of models: (∞, 1)-categories
in which we can interpret the axioms of our type theory. Studying various models can help us
discern which statements can and cannot be proven with our given axiomatization.
Due to a result by Shulman, we already know that every Grothendieck (∞, 1)-topos is a
model for homotopy theory. However, we do not expect all models to be a Grothendieck (∞, 1)-
topos and in particular, we anticipate non-presentable models of homotopy type theory.
The goal of this work is to take a first step towards showing the existence of non-presentable
models of homotopy type theory, by constructing a non-presentable elementary (∞, 1)-topos
. Elementary (∞, 1)-toposes share many features with Grothendieck (∞, 1)-toposes (such as
descent, universes, natural number objects, ... ), but are not required to be presentable and thus
can include examples that are not Grothendieck (∞, 1)-toposes.
We will construct such examples via the filter construction. Generalizing a result from 1-
category theory, we prove that for every elementary (∞, 1)-topos E and filter of subobjects
Φ, we can construct an elementary (∞, 1)-topos Φ E, which is in fact not presentable if the
filter Φ is not principal. We will apply this result to the (∞, 1)-category of Kan complexes to
construct non-presentable examples of elementary (∞, 1)-toposes.
636
