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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The magic of the morphisms

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, lets call it a spotted set in analogy with pointed sets. Given two spotted sets, then a morphism $\...
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If functions compose both ways to make automorphisms, are they isomorphisms?

Let's say that we have morphisms $f:A \to B$ and $g : B \to A$ such that $f \circ g$ and $g \circ f$ are both automorphisms (an automorphism is a morphism that is both iso and endo). Are $f$ and $g$ ...
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Extension of projection from a point to a Blow Up

I feel like there's something obvious I'm missing here, and I'm not looking for a whole answer, but rather just a pointer in the right direction. Suppose you have the projection from a point $\mathbb{...
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Morphism between affine scheme corresponding with ring homomorphism

Let $X = \text{Spec} R$ and $K = \text{Spec} S$ be two affine schemes with $f,g: K \rightarrow X$. I know that a morphism between affine schemes correspond with a ring homomrphism, so denote $\varphi: ...
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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 ...
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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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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 ...
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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 ...
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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 Gröbner degeneration from an ideal I to its ...
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If $\{v_1, v_2, …, v_n\}$ is a basis and $f$ is an injective morphism, show that $\{f(v_1), f(v_2), …, f(v_n)\}$ is linearly independent.

Let $V_1$, $V_2$ be two $K$-vector spaces with $dim_K V_1 = dim_K V_2 = n$, $f:V_1 \rightarrow V_2$ a morphism and $B = \{v_1, v_2, ..., v_n\}$ a basis for $V_1$. Now consider the set $T = \{f(v_1), ...