# Questions tagged [morphism]

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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### If $\mathcal{C}$ has pullbacks & coequalisers, and the pullback of a regular epi is epic, morphisms factorise as regular epic & monic

I'm trying to show that if $\mathcal{C}$ has pullbacks & coequalisers, and the pullback of a regular epimorphism is epic, morphisms factorise as a regular epic followed by a monic. My approach so ...
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### Is there example of equalizer that is not injective?

we know that in general case every arrow is not function also in the $Div \mathbb{A}b$ there is example of monic that is not injective. Is there example of equalizer that is not injective ?
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### Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds?

Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds ? In other words, isomorphism between algebraic smooth curves has always degree 1 ? Degree of morphism ...
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### Initial objects in the category of bilinear maps $M \times N \to L$. where $M,N$ are fixed $R$-modules, and $L$ arbitrary $R$-module.

In our class we are looking at the definition of tensor product, and here’s a (paraphased) remark that I don’t get. Let $M,N$ be fixed $R$-modules, and $L$ an arbitrary $R$-module. At this point we ...
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### What is an isomorphism between $L:aX+bY=Z$ and $\Bbb P^1_K$?

It is known that genus $0$ smooth curve over field $K$ with base point is isomorphic to $\Bbb P^1_K$ over $K$. I want to understand this with a lot of examples. For example, let $a,b\in K^\times$ and ...
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### A more succinct group object diagram (all axioms in one connected diagram), questions about its properties...

Here is the definition of group object from nLab. They give 3 associated maps $* \xrightarrow{1} G$, $m: G^2 \to G$, and $-^{-1}: G \to G$ and require 3 commutative diagrams to complete the axioms ...
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Take $A$ a commutative unitary $\mathbb{C}$-algebra and take $A_0\subset A_1\subset...$ a filtration on $A$. If $$GrA=\bigoplus \frac{A_n}{A_{n-1}}$$with $A_{-1}=0$, is the associated graded algebra. ...