27 research outputs found
Multitensor lifting and strictly unital higher category theory
In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result -- the lifting theorem for multitensors -- enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak n-categories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)-categories with strict units
The homotopy theory of simplicial props
The category of (colored) props is an enhancement of the category of colored
operads, and thus of the category of small categories. In this paper, the
second in a series on "higher props," we show that the category of all small
colored simplicial props admits a cofibrantly generated model category
structure. With this model structure, the forgetful functor from props to
operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat
The groupoidal analogue Theta~ to Joyal's category Theta is a test category
We introduce the groupoidal analogue \tilde\Theta to Joyal's cell category
\Theta and we prove that \tilde\Theta is a strict test category in the sense of
Grothendieck. This implies that presheaves on \tilde\Theta model homotopy types
in a canonical way. We also prove that the canonical functor from \Theta to
\tilde\Theta is aspherical, again in the sense of Grothendieck. This allows us
to compare weak equivalences of presheaves on \tilde\Theta to weak equivalences
of presheaves on \Theta. Our proofs apply to other categories analogous to
\Theta.Comment: 41 pages, v2: references added, Remark 7.3 added, v3: metadata
update
Homotopy Theoretic Models of Type Theory
We introduce the notion of a logical model category which is a Quillen model
category satisfying some additional conditions. Those conditions provide enough
expressive power that one can soundly interpret dependent products and sums in
it. On the other hand, those conditions are easy to check and provide a wide
class of models some of which are listed in the paper.Comment: Corrected version of the published articl
Group actions on Segal operads
We give a Quillen equivalence between model structures for simplicial
operads, described via the theory of operads, and Segal operads, thought of as
certain reduced dendroidal spaces. We then extend this result to give an
Quillen equivalence between the model structures for simplicial operads
equipped with a group action and the corresponding Segal operads.Comment: Revised version. Accepted to Isr J Mat
The de Rham homotopy theory and differential graded category
This paper is a generalization of arXiv:0810.0808. We develop the de Rham
homotopy theory of not necessarily nilpotent spaces, using closed dg-categories
and equivariant dg-algebras. We see these two algebraic objects correspond in a
certain way. We prove an equivalence between the homotopy category of schematic
homotopy types and a homotopy category of closed dg-categories. We give a
description of homotopy invariants of spaces in terms of minimal models. The
minimal model in this context behaves much like the Sullivan's minimal model.
We also provide some examples. We prove an equivalence between fiberwise
rationalizations and closed dg-categories with subsidiary data.Comment: 47 pages. final version. The final publication is available at
http://www.springerlink.co
Lifting and restricting recollement data
We study the problem of lifting and restricting TTF triples (equivalently,
recollement data) for a certain wide type of triangulated categories. This,
together with the parametrizations of TTF triples given in "Parametrizing
recollement data", allows us to show that many well-known recollements of right
bounded derived categories of algebras are restrictions of recollements in the
unbounded level, and leads to criteria to detect recollements of general right
bounded derived categories. In particular, we give in Theorem 1 necessary and
sufficient conditions for a 'right bounded' derived category of a differential
graded(=dg) category to be a recollement of 'right bounded' derived categories
of dg categories. In Theorem 2 we consider the particular case in which those
dg categories are just ordinary algebras.Comment: 29 page