# Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

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### Universal property definition of an ideal generated by a subset?

I'm puzzled by the definition of ideals generated by a subset of a ring in Aluffi, Algebra: Ch 0. The previous chapter on groups is (for an algebra book of this level) quite categorical in spirit. In ...
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### Universal property of HNN extensions

The amalgamated free product of groups can be defined as a certain presentation; or, one can define it as a pushout where the morphisms are embeddings. I was wondering if the definition of an HNN ...
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### Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions: Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
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### Using only the universal property, prove that $X$ is dense in its Cauchy completion

Long ago, I learned about the Cauchy completion of metric spaces via the usual explicit construction of quotienting the set of Cauchy sequences. For a metric space $X$, let $\hat X$ denote this Cauchy ...
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### On the universal property of Tensor Product

I was readying the construction of the Tensor Product that was made on the book Introduction to Smooth Manifolds, from John M. Lee. In the proposition 12.7 he proved that his construction satisfied ...
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### Difference betwen universal properties

I'm studying the construction of the tensor product. To do this, I opted for the quotient space's approach, defining first the free vector space over the cartesian $V\times W$ of two vector spaces. ...
1 vote
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### Reference for quotient lattices and universal property?

Question: Are there any references explaining the definition (or definitions) of quotient objects in the category of lattices? In particular a characterization in terms of universal properties would ...
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### Equivalence statement for existence product of modules

The following exercise is taken from T.S. Blyth book, Module Theory, chapter 6, exercise 9: Something does not add up, especially with using (1). It seems that condition (1) is not necessary or ...
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### Understanding the Universal Property of Tensor Products for Modules Over Any Ring

I am delving into the concept of tensor products and their universal properties in the context of modules over rings. While the universal property of tensor products is well-established for vector ...
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1 vote
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### How do I prove $U(\mathfrak{g} \oplus \mathfrak{h})\cong U(\mathfrak{g})\otimes U(\mathfrak{h})$ using the universal properties?

Reading lecture notes on Hopf algebras, I came across a statement (which was heavily used by the author without a proof) that, given Lie algebras $\mathfrak{g}, \mathfrak{h}$, the universal enveloping ...
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### Is universal object unique up to unique isomorphism?

If $F:C \to D$ is functor between categories $C$ and $D$ and $X$ is an object of $D$, then a universal map from $X$ to $F$ is a pair $(A_X,u)$ where $A_X \in \mathrm{ob}(C)$ and $u \in D(X,F(A_X))$ ...
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### Does a product need to be unique up to unique isomorphism?

I’m re-reading Bartosz Milewski’s Category Theory for Programmers and I am a bit confused. As I understand it, the universal construction (apparently called universal property elsewhere) is unique up ...
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### Categorical notion of finiteness

TLWR : I'd like to define either $\operatorname{Hom}_F : \text{Set} \times \text{Set}_* \to \text{Set}_*$ (set of functions with finite support) or simply $\mathcal{P}_F : \text{Set} \to \text{Set}$ (...
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### Proving that 2 formulations of the Universal Property of Free Groups are equivalent

The present question is inspired by an answer to this question of mine on applications of Category Theory in Abstract Algebra. One of the answers stated that the universal property of free groups ...
$\def\codom{\operatorname{codom}}$If $G$ is a first-countable abelian topological group, one can find a morphism $G\to\hat{G}$ of abelian topological groups, with $\hat{G}$ first-countable and ...