1,998 research outputs found
Higher quasi-categories vs higher Rezk spaces
We introduce a notion of n-quasi-categories as fibrant objects of a model
category structure on presheaves on Joyal's n-cell category \Theta_n. Our
definition comes from an idea of Cisinski and Joyal. However, we show that this
idea has to be slightly modified to get a reasonable notion. We construct two
Quillen equivalences between the model category of n-quasi-categories and the
model category of Rezk \Theta_n-spaces showing that n-quasi-categories are a
model for (\infty, n)-categories. For n = 1, we recover the two Quillen
equivalences defined by Joyal and Tierney between quasi-categories and complete
Segal spaces.Comment: 44 pages, v2: terminology changed (see Remark 5.27), Corollary 7.5
added, appendix A added, references added, v3: reorganization of Sections 5
and 6, more informal comments, new section characterizing strict n-categories
whose nerve is an n-quasi-category, numbering has change
A Quillen model structure for Gray-categories
A Quillen model structure on the category Gray-Cat of Gray-categories is
described, for which the weak equivalences are the triequivalences. It is shown
to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to
provide a functorial and model-theoretic proof of the unpublished theorem of
Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model
structure on Gray-Cat is conjectured to be Quillen equivalent to a model
structure on the category Tricat of tricategories and strict homomorphisms of
tricategories.Comment: v2: fuller discussion of relationship with work of Berger;
localizations are done directly with simplicial set
Rational Combinatorics
We propose a categorical setting for the study of the combinatorics of
rational numbers. We find combinatorial interpretation for the Bernoulli and
Euler numbers and polynomials.Comment: Adv. in Appl. Math. (2007), doi:10.1016/j.aam.2006.12.00
A cartesian presentation of weak n-categories
We propose a notion of weak (n+k,n)-category, which we call
(n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant
objects of a certain model category structure on the category of presheaves of
simplicial sets on Joyal's category Theta_n. This notion is a generalization of
that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces).
Our main result is that the above model category is cartesian.Comment: Incorporates corrections to the published version, which appeared in
a separate correction not
Shadows and traces in bicategories
Traces in symmetric monoidal categories are well-known and have many
applications; for instance, their functoriality directly implies the Lefschetz
fixed point theorem. However, for some applications, such as generalizations of
the Lefschetz theorem, one needs "noncommutative" traces, such as the
Hattori-Stallings trace for modules over noncommutative rings. In this paper we
study a generalization of the symmetric monoidal trace which applies to
noncommutative situations; its context is a bicategory equipped with an extra
structure called a "shadow." In particular, we prove its functoriality and
2-functoriality, which are essential to its applications in fixed-point theory.
Throughout we make use of an appropriate "cylindrical" type of string diagram,
which we justify formally in an appendix.Comment: 46 pages; v2: reorganized and shortened, added proof for cylindrical
string diagrams; v3: final version, to appear in JHR
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