Whether the morphism has special name

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In summary, if a morphism h is an extension of a morphism f, then f is considered a restriction of h. Similarly, if h is a lifting of f, the term for f in relation to h depends on how it was lifted. There can be multiple lifts without one being considered the "correct" one. An example where a special property exists is the induced morphism in representation theory, which requires a functor for existence, uniqueness, and the correct universal properties.
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If morphism h is an extension of morphism f,we call morphism f is a restricion of morphism h; If morphism h is a lifting of morphism f,what do we call morphism f of morphism h ? Or in other words, if i is an inclusion morphism ,then what do we call the morphism f*i, as i*f is called restriction of f ?
 
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What you call it entirely depends on how one has lifted; there are normally going to be (infinitely) many different lifts with no reason to suppose that anyone is _the_ lift.

An example where one does have some special properties would be the induced morphism in representation theory from the induction functor. (One might well need some functor lying around to show that something exists and is unique and has the right universal properties.)
 

1. What is a morphism?

A morphism is a mathematical concept that describes the relationship between two mathematical objects or structures. It is typically represented as a function or mapping that preserves certain properties between the objects.

2. What makes a morphism special?

A morphism can be considered special if it has certain characteristics or properties that make it unique or useful in a particular context. These properties can vary depending on the specific field of mathematics being studied.

3. What are some examples of named morphisms?

Some commonly known named morphisms include isomorphisms, homomorphisms, and endomorphisms. Isomorphisms preserve structure between objects, homomorphisms preserve operations, and endomorphisms map an object to itself.

4. How are morphisms named?

In mathematics, morphisms are typically named based on their properties or the objects they relate. For example, a group homomorphism relates two groups, and a ring isomorphism relates two rings. In some cases, they may also be named after the mathematician who discovered or studied them.

5. Why are morphisms important?

Morphisms play a crucial role in abstract algebra and category theory, which are fundamental branches of mathematics. They allow for the comparison and study of different mathematical objects and structures, and can reveal deeper connections and patterns between them.

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