# Tagged Questions

In category theory, a morphism is a structure-preserving map, such as continuous mappings on topological spaces, measurable functions, and linear maps.

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### Are there two groups $G_1 , G_2$ of orders $7$ and $8$ respectively & a morphism $f: G_1 \to G_2$ such that $|\operatorname{Im}(f)| = 4$?

Is is possible to find two groups $G_1, G_2$ of orders $7$ and $8$ respectively & a morphism $f: G_1 \to G_2$ such that $|\operatorname{Im}(f)| = 4$ ?
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### Why do we study the number of homomorphisms/isomorphisms between fields?

From the first abstract algebra class, we encounter many problems that ask us to find the number of homomorphisms/isomorphism a between two algebra structures (e.g. field). My first question is why ...
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### Gröbner degeneration, flat families and morphism of varieties.

I am trying to understand why a map is actually a morphism of varieties. The map, from the affine line to the Hilbert scheme of points, is given by a GrÃ¶bner degeneration from an ideal I to its ...
0answers
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### Functions which map Lebesgue measurable sets into Lebesgue measurable sets

Can you give an alternative description of the set of functions which map Lebesgue measurable sets into Lebesgue measurable sets? I think that it is enough to consider Lebesgue measurable sets on the ...
1answer
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### What is the link between homomorphisms and mutual information?

Intuitively, there seems to be a link between the (kind of) homomorphism between two algebraic structures and the mutual information between two variables. However, since I'm not a mathematician, it's ...
1answer
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### Why are “the” morphisms of the category of topological spaces continuous maps?

On the Wikipedia page for "morphism (category theory)" it says that: In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms. In what ...
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1answer
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### Are Monomorphisms injective?

In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective. Things are different for schemes. In fact I know that a scheme injective ...