# A MODEL STRUCTURE ON INTERNAL CATEGORIES IN SIMPLICIAL SETS

@article{Horel2015AMS, title={A MODEL STRUCTURE ON INTERNAL CATEGORIES IN SIMPLICIAL SETS}, author={Geoffroy Horel}, journal={arXiv: Algebraic Topology}, year={2015} }

We put a model structure on the category of categories internal to sim- plicial sets. The weak equivalences in this model structure are preserved and reected by the nerve functor to bisimplicial sets with the complete Segal space model struc- ture. This model structure is shown to be a model for the homotopy theory of innity categories. We also study the homotopy theory of internal presheaves over an internal category.

## 15 Citations

### Rigidification of higher categorical structures

- Mathematics
- 2015

Given a limit sketch in which the cones have a finite connected base, we show that a model structure of "up to homotopy" models for this limit sketch in a suitable model category can be transferred…

### Univalence for inverse EI diagrams

- Mathematics
- 2015

We construct a new model category presenting the homotopy theory of presheaves on "inverse EI $(\infty,1)$-categories", which contains universe objects that satisfy Voevodsky's univalence axiom. In…

### A note on the (∞,n)–category of cobordisms

- MathematicsAlgebraic & Geometric Topology
- 2019

In this note we give a precise definition of fully extended topological field theories a la Lurie. Using complete n-fold Segal spaces as a model, we construct an $(\infty,n)$-category of…

### An Introduction to Higher Categories

- MathematicsAlgebra and Applications
- 2019

In this chapter we give a non-technical introduction to higher categories. We describe some of the contexts that inspired and motivated their development, explaining the idea of higher categories,…

### Segal objects and the Grothendieck construction

- Mathematics
- 2016

We discuss right fibrations in the $\infty$-categorical context of Segal objects in a category V and prove some basic results about these.

### Brown categories and bicategories

- Mathematics
- 2015

In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we show how to assemble these spaces into…

### All $(\infty,1)$-toposes have strict univalent universes

- Mathematics
- 2019

We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type…

### Segal-type models of higher categories

- Mathematics
- 2017

Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key…

### Factorization Homology as a Fully Extended Topological Field Theory

- Mathematics
- 2014

Given an En-algebra A we explicitly construct a fully extended n-dimensional topological field theory which is essentially given by factorization homology. Under the cobordism hypothesis, this is the…

## References

SHOWING 1-10 OF 30 REFERENCES

### A model category structure on the category of simplicial categories

- Mathematics
- 2004

In this paper we put a cofibrantly generated model category struc- ture on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of…

### Simplicial Homotopy Theory

- MathematicsModern Birkhäuser Classics
- 2009

Simplicial sets, model categories, and cosimplicial spaces: applications for homotopy coherence, results and constructions, and more.

### Partial model categories and their simplicial nerves

- Mathematics
- 2011

In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More…

### A model for the homotopy theory of homotopy theory

- Mathematics
- 1998

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more…

### A characterization of simplicial localization functors and a discussion of DK equivalences

- Mathematics
- 2012

### Model structures on the category of small double categories

- Mathematics
- 2007

In this paper, several model structures on DblCat, the category of small double categories, are obtained and explicit descriptions for and discuss properties of free double Categories, quotient double category, colimits of double categories), horizontal nerve and horizontal categorification are given.

### Homotopical resolutions associated to deformable adjunctions

- Mathematics
- 2014

Given an adjunction FaG connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the…