Maths - Concrete Category - Topology

There are various categories associated with topology. The most fundamental is Top - Category of Topological Spaces. There is a category of the subset structure of open sets. There are also categories of fibres and sheaves.

Top - Topological Spaces

Consider a category C as follows:

For a non-category theory view of this structure see page here.

Subset Structure of Open Sets as Category

Subsets give interesting structure to open sets. This allows us to define a category of topological spaces. We can also further elaborate this subset structure to get fibre bundles and sheaves.

In this diagram the open set U is a subset of V.

UcontainsV

We can represent this as an arrow from V to U

V->U

topological space

This has the properties we expect from a category, for instance, the identity map:

U->U (identity map)

topological space

and composition

(U->V)*(V ->W) = U->W (composition)

topological space

For a non-category theory view of this structure see page here.

Bn - Category of Bundles

A fibre bundle is a function f:(A->I) This is described on the following pages:

We can make this a category where:

  • objects are pairs (A,f) consisting of a stalk 'A' and a mapping of 'A' to 'I'.
  • morphisms are functors between these where I does not change.

This triangle must commute. So the elements (germs) of a stalk in 'A' must map to the same stalk in 'B'.

This is a comma category as discussed on this page.

The Bn Category can lead on to the concept of a Topos.

For more information about the Bn category see this page.

Simplical Sets

We have looked at these, mostly from a topology point of view, on the pages here:

The subject can also be approached from a purely combinatorial point of view.

Here we investigate how these structures can be viewed in a category theory way.

Δ is a category with:

  • objects - are ordered sets which represent simplicies.
  • morphisms - are order preserving maps.

The morphisms are inclusions

Δop is a category with the same objects but morphisms are the face maps.

 

Category of Presheaves

Objects: Cop -> set

contraveriant functors X: C -> set

(written X: Cop -> set to indicate contraveriance)

category of presheaves

Morphisms:

are natural transformations N: X -> Y

morphisms in presheaf

For a non-category theory view of this structure see page here.


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