Problem 1 - Algebraic Topology

Question

Show that \(\operatorname{Ext}(H, G)\) is a contravariant functor of \(H\) for fixed \(G,\) and a covariant functor of \(G\) for fixed \(H.\)

Step-by-Step Solution

Answer

\(\operatorname{Ext}(H, G)\) reverses homomorphisms for \(H\) (contravariant) and maintains them for \(G\) (covariant), meeting functorial conditions.

Step 1: Understanding the Functor Concept

Since \(\operatorname{Ext}(H, G)\) is a derived functor, its contravariant and covariant properties are tied to how it transforms under homomorphisms.A contravariant functor reverses the direction of morphisms, while a covariant one maintains it.

Step Folding the Problem for Contravariance

To show that \(\operatorname{Ext}(H, G)\) is contravariant in \(H\), consider homomorphisms \(f: H' \to H\). This induces a reversed map \(\operatorname{Ext}(H, G) \to \operatorname{Ext}(H', G)\).

Step Folding the Problem for Covariance

For covariance in \(G\), consider homomorphisms \(g: G \to G'\). The induced map is \(\operatorname{Ext}(H, G) \to \operatorname{Ext}(H, G')\), retaining the morphism direction.

Step Validating with Functorial Properties

We need to validate functorial properties like identity and composition: - For identity, the maps \(\operatorname{Ext}(H, G) \to \operatorname{Ext}(H, G)\) remain unchanged. - For composition, given morphisms \(f: H'' \to H'\) and \(f': H' \to H\), it ensures \(\operatorname{Ext}(H'' , G) \to \operatorname{Ext}(H' , G) \to \operatorname{Ext}(H, G)\) and vice versa for covariant cases.

Step 5: Conclusion of Functorial Proving

These transformations satisfy the requirements of a contravariant functor in \(H\) and a covariant functor in \(G\).

Key Concepts

Contravariant Functor

Understanding the contravariant functor involves recognizing how this type of functor interacts with categories and morphisms. In category theory, a functor is a mapping between categories that preserves the structure of those categories. A contravariant functor, however, has the interesting property of reversing the direction of morphisms.
Contravariant functors essentially map each morphism in the opposite direction, which can seem counterintuitive. But consider it as flipping the arrows in a path that relates the objects in two categories.
For example, in the context of the Ext functor, when dealing with a fixed group, morphisms between groups will typically flow from one group to another. For a fixed G in the Ext functor, \[\text{Ext}(H', G) \rightarrow \text{Ext}(H, G)\] shows that the morphism is reversed when transitioning from H to a related H'. This property is crucial for various mathematical situations where reversing morphism is necessary to fit a model or solve an equation.

Covariant Functor

Unlike a contravariant functor, a covariant functor preserves the direction of morphisms as it transfers categories or structures. This intuitive approach aligns with our natural understanding of mappings. When a covariant functor transforms objects and morphisms from one category to another, it maintains directionality.
In \[\text{Ext}(H, G) \rightarrow \text{Ext}(H, G')\] for fixed H, given any morphism g from G to G', the direction of transformation is preserved. Here, we notice there's no change in how arrows, or morphisms, maintain orientation. Working with covariant functors can be simpler since they behave in a "normal" way, maintaining the forward flow of morphisms, which is directly aligned with most mathematical operations.
This property is fundamental because it allows for straightforward composition of functions or transformations without requiring adjustments for direction, thereby easing many algebraic manipulations and proofs.

Functorial Properties

Functorial properties provide the backbone for understanding how functors interact with morphisms and objects within categories. There are specific properties that functors must satisfy to be correctly regarded as either contravariant or covariant.

• Identity Preservation: A functor, irrespective of its variance, must map identity morphisms to identity morphisms. This means if you have an identity morphism on an object in a category, the functor should map it to an identity morphism on the image of that object in the other category.


• Composition Preservation: Functors also need to preserve the composition of morphisms. For covariant functors, if you have two morphisms \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\), the functor maps the composition \(g \circ f\) to \(F(g) \circ F(f)\). For contravariant functors, the composition direction is reversed, ensuring the order \(F(f) \circ F(g)\).

These properties ensure meaningful and consistent ways to map structures from a category to another, supporting the foundational concepts of category theory.