# Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

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### Formalizing the universal property of the tensor product

Introduction (a bit long) I am very new to category theory and algebra, in general. I recently saw the formalized concept of a universal morphism, here in WikipediA. I will repeat the definition with ...
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### Universal property of the (Vistoli-)sheafification

Given a presheaf $F$, in Notes on Grothendieck topologies, fibered categories and descent theory there is a construction of the sheafification (Proof for theorem 2.64). The first part is the ...
1 vote
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### Missing universal properties

Given a category $\mathcal{C}$, we can use copresheaves $H \in [\mathcal{C},Set]$ or presehaves $H \in [\mathcal{C}^{\mathrm{op}},Set]$ to state left or right universal properties. Existence of a ...
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### Well-definedness of tensor product of modules

Given two $R$-modules $M$ and $N$, one possible construction of the tensor product $M\otimes_R N$ is by taking the quotient $F/U$ of the free module $F=\bigoplus_{m,n}R\cdot(m,n)$ and the submodule $U$...
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### Can fibrations be described via universal morphisms?

Jacobs defines Cartesian morphisms and fibrations in Definition 1.1.3 of this document. In other places in the text he uses phrases like "the universal property of this lifting." This made ...
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### Does the universal set operate as the Von Neumann universe in non well founded set theories?

I've been researching non well founded set theories (E.g. NF, NFU, etc.) and have been wondering if there are any similarities between the universal set & Von Neumann universe ? Or if there the ...
1 vote
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### Universal Property of The Simplex Category Δop

I tried formulating the universal property of the simplex category. There seemed to be two different options to choose for the universal property of $\Delta^{op}$. I'm having trouble making ...
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### Kan extensions as weighted limits

In this Blog David Jaz Myers says that right Kan extensions can be expressed using weighted limits. I would really appreciate any help understanding how this is so. I do not understand the sense in ...
1 vote
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### Why $\bar{f}(x + H_1) = f(x) + H_2$?

I am trying to solve part $(c)$ of the following problem: For $i = 1,2,$ let $G_i$ be a group and $H_i$ be a normal subgroup of $G_i.$ Let $\pi_i: G_i \to G_i/H_i$ be given by $\pi_i(a) = a H_i$ for ...
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### Prove: there exist a unique group homomorphism $\bar{f}: G_1/H_1 \to G_2/H_2$

I am trying to solve the following problem: For $i = 1,2,$ let $G_i$ be a group and $H_i$ be a normal subgroup of $G_i.$ Let $\pi_i: G_i \to G_i/H_i$ be given by $\pi_i(a) = a H_i$ for all $a \in G_i.$...
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### Why do we require uniqueness in the universal property for a fraction field?

I'm a university student taking a course in abstract algebra. My professor recently introduced fraction fields, giving this definition: Let $R$ be an integral domain. There exists a field $F$, called ...
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### Finite product and coproduct are the same in abelian categories

I am working with the following definition of abelian category. a) It has a $0$ object. b) Every morphism has a kernel and a cokernel. Every monomorphismis a kernel and every epimorphism is a ...
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### Q: Does stictly increasing function always cross x and y axis?

I've been trying to figure out this problem: Let f: be strictly increasing function on it's Domain with range of all real numbers. Does it always intercept x and <...
Let $G$ be a graph. We say that a category $\textbf{C}$ has a universal mapping property of free categories on $G$ if there exists a graph homomorphism $i:G\longrightarrow |\textbf{C}|$ such that ...