# Is Hom an additive functor?

## Is Hom an additive functor?

The hom-functor Hom(−,−):𝒜op×𝒜→Ab is additive (and in both arguments separately).

**Is the Hom functor exact?**

−→ Hom(M,A). In other words, the functor Hom(M,−) is left-exact and sends kernels to kernels.

### Is Hom functor faithful?

In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category SetCop (covariant or contravariant depending on which Hom functor is used).

**What is Hom in algebra?**

A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism.

#### What is an additive functor?

Additive functors A functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams.

**What is Hom-set?**

hom-set (plural hom-sets) (category theory) The set or collection of all morphisms from A to B for some given ordered pair (A, B) of objects from some given category.

## What is the meaning of morphism?

The form –morphism means “the state of being a shape, form, or structure.” Polymorphism literally translates to “the state of being many shapes or forms.” What are some words that use the combining form –morphism? allomorphism.

**How do functors work?**

In other words, a functor is any object that can be used with () in the manner of a function. This includes normal functions, pointers to functions, and class objects for which the () operator (function call operator) is overloaded, i.e., classes for which the function operator()() is defined.

### What is hom a B?

Noun. hom-set (plural hom-sets) (category theory) The set or collection of all morphisms from A to B for some given ordered pair (A, B) of objects from some given category.

**What is Hom set?**

#### Why is it called a hom set?

“Hom” stands for homomorphism, the usual name for structure preserving functions in algebra. I believe the terminology goes back to Eilenberg and Mac Lane’s original article on category theory. Of course, in an arbitrary category the objects need not have structure, and the morphisms need not even be functions.

**What is a linear category?**

A linear category, or algebroid, is a category whose hom-sets are all vector spaces (or modules) and whose composition operation is bilinear. This concept is a horizontal categorification of the concept of (unital associative) algebra.