# Cellular objects in the motivic model category

Friday, November 16th, 2012 | Author:

This article explains the notion of cellular objects in the motivic model category, which are like CW complexes for the algebraic/motivic world.

# Cellular objects: CW complexes

Wednesday, November 14th, 2012 | Author:

We will investigate the notion of cellular objects in a model category; today: the classical case of CW-complexes in the model category of topological spaces with Serre-fibrations as fibrations.

A CW complex is a certain kind of topological space, together with a CW structure, which is a description how to glue the space from spheres (or from affine spaces, if you prefer), called the cells. The acronym CW stands for "closure finite, weak topology", which I will explain soon. CW complexes are a class of spaces broader than simplicial complexes, and they are still combinatorial in nature.

# Model categories

Monday, November 12th, 2012 | Author:

This is supposed to be a short intuitive introduction to model categories.

Suppose you have a category $\mathcal{C}$ and some class of morphisms $W$ which behave somewhat like isomorphisms (for example: Chain complexes and Quasi-isomorphisms, or topological spaces and homotopy equivalences, or simplicial sets and weak homotopy equivalences ...). We will call this class "weak equivalences". Then you can look at the localized category $[W^{-1}]\mathcal{C}$, where the morphisms in $W$ are made invertible. If you're lucky, not all objects are isomorphic to each other, and if you're really lucky, you can compute something.

But, as it turns out, usually you don't work with the localized category abstractly, but by some explicit construction of some special case (say, Verdier localization of triangulated categories in the homological setting or explicit homotopies in the topological setting).

Model categories (and its cousins, weak factorization systems, categories of (co)fibrant objects, homotopical categories, etc.) provide a framework to compute stuff in $[W^{-1}]\mathcal{C}$.

[UPDATE 2013-03-06] I gave a 30-Minute talk about model categories, with very little content. [/UPDATE]

Category: English, Mathematics | 2 Comments

# Homotopy limits

Tuesday, November 06th, 2012 | Author:

In this short posting, I want to give some intuitive idea on homotopy limits. Homotopy (co)limits appear whenever one has a notion of homotopy equivalence or weak equivalence between objects and one doesn't want to have constructions that distinguish between equivalent objects. The most prominent settings are, of course, classical homotopy theory and homological algebra. Although not necessary for the definition of homotopy (co)limits, I also talk about model categories.

# What is ... a vector bundle?

Thursday, November 01st, 2012 | Author:

A vector bundle is a morphism that looks locally on the target like a product of the target with a vector space.

We will call the target space the base and the space of definition the total space. The preimage of a point of the base is called the fiber.

Is that the correct mathematical definition? It doesn't mention what kind of spaces we look at, what kind of morphism I'm talking about, what the product is, locally in which sense, vector space over which field, do we allow infinite dimension, ... so it's not a mathematical definition in the pedantic sense. I will give you pedantic definitions in this article, just to satisfy my need to write down what I consider to be a good terminology.

Nowadays it is common to use $x \mapsto f(x)$ to denote that an element $x \in X$ is mapped to an element $f(x) \in Y$ by the map(ping) $f : X \to Y$. In particular, the arrow $\rightarrow$ (in LaTeX: \rightarrow) denotes a map, or more generally a morphism, while $\mapsto$ (in LaTeX: \mapsto) denotes how particular elements or objects are mapped to other elements or objects.