# Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

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### What are colored morphisms/arrows intended to mean in these diagrams?

I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include ...
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### Looking for properties of a pseudo-logarithm where you add all the (prime factor$\cdot$exponent) terms in an integer's prime factorization?

If, for positive integer $n=p_1^{q_1}p_2^{q_2}\dots p_k^{q_k}$ with the right side the prime factorization of $n$, we define $J(n)$ as $p_1q_1+p_2q_2+\dots+p_kq_k$, does $J(n)$ have any generally &...
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### Is this diagram of the free product of two groups correct?

From Wikipedia: "In mathematics, specifically group theory, the free product is an operation that takes two groups $G$ and $H$ and constructs a new group $G ∗ H$. The result contains both $G$ and ...
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### Wikipedia definition of Image in Category theory applied to Set

According to Wikipedia, the image of a morphism $f:A \to B$ is a monomorphism $m:I \to B$ for which there is some $e:A \to I$ with $f=m\circ e$, and such that the following universal property is ...
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### Localising a ring twice gives ring isomorphic to localising by each subset in turn

Given a ring $R$, suppose we have two multiplicative subsets $S,U \subseteq R$ (which contain 1). Write $US = \{us \mid u \in U, s \in S\}$ also a multiplicative subset, containing both $S$ and $U$ ...
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### How to derive this from the universal property of the ring of fractions?

(All the involved rings are commutative with identity.) The "universal" definition that I follow is this: Let $S$ be a multiplicative subset of a ring $A$ which also contains $1_A$. Then a ...
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### Does the direct sum have a universal property in the category of groups?

In other categories, like modules, the direct sum is the coproduct, giving it a neat category theory description. The direct product of groups, which is very similar to it, is the product in the ...
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### In category theory, does a definition via a universal property induce a functor?

In category theory one frequently defines concepts via universal properties. Examples are (co)equalizers of pairs of morphisms having the same domain and codomain (co)kernels of morphisms in ...
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### Question about the meaning of a "region" in mathemathics.

I am currently going over double integral. One concept that seems to appear a lot is "region". I have consulted different resources and it seems like there is no objetive definition of what ...
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### Topical example of a universal arrow to and from a functor

If $S\colon \mathsf D\to \mathsf C$ is a functor between categories and $c\in \mathsf C$, Mac Lane in his Categories for the Working Mathematician defines a universal arrow from $c$ to $S$ as ...
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### AF-Algebras and Matrix Algebras

I have been having some trouble with an exercise from Murphy's "$C^*$-Algebras and Operator Theory" recently. Chapter 6 Exercise 6 is as follows: If $A$ is an AF-Algebra (i.e. a direct limit ...
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### Embedding a free algebra in a product.

Let $A$ be an algebra; set $V=HSP(A)$. Then for any $X\neq\emptyset$, there is an embedding of the free algebra $F_{V}(X)$ into the algebra $A^{A^{X}}$, $\Psi\colon F_V(X)\rightarrow A^{A^X}$ given by ...
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### Has this constructible property of structures in model theory been studied?

I was looking into model theory where a first-order language can be interpreted with a structure and a variable assignment. Here is an example of a FOL with interpretation: We have terms: Atomic ...
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### Uniqueness of the morphism in the universal property

If we want to define a product of objects in a category (cf. Wikipedia Page), we need to find a morphism and it must be unique. In our notes, the professor gives some examples such as the product of ...
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### Characterizations of product topology and box topology

Given a collection of topological spaces $X_i$ indexed by the elements $i$ of a set $I$, we consider the set product $P = \prod_{i \in I} X_i$ with projections $p_i : P \to X_i$. There are two methods ...
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### Convenience of using universal property of tensor product

On the wikipedia page of universal property, it says one of the motivations of universal property is The concrete details of a given construction may be messy, but if the construction satisfies a ...
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### Direct limit characterized up to isomorphism

This is from Atiyah & Macdonald - Exercise 2.16: We are given a direct system $(M_i,\mu_{ij})$ with direct limit $M=\underset{\longrightarrow}{\lim} M_i$. Show that the direct limit is ...
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### Show that two tensor products are isomorphic.

Let $M$ and $N$ be two $R$-modules,then a pair $(T,\varphi)$ where $T$ is an $R$-module and $\varphi:M\times N\to T$ is a bilinear map is called a tensor product if for any pair $(P,f)$ where $P$ is ...
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Let $(X,\mathscr M,\mu)$ be a measure space. The completion of $(X,\mathscr M,\mu)$ is the measure space $(X,\bar{\mathscr M},\bar\mu)$ where $$\bar{\mathscr M} = \{{E\cup F : \text{ E\in \... 1 vote 1 answer 64 views ### If R is an integral domain, then Frac(R[x_1,…,x_n])= Frac(K[x_1,…,x_n]) where K=Frac{(R)} I’m trying to show that if R is an integral domain, then Frac(R[x_1,…,x_n])= Frac(K[x_1,…,x_n]) where K=Frac{(R)} (here Frac{(R)} denotes “field of fractions of R”). I think I can use the ... 2 votes 1 answer 175 views ### Complexification of O(n,\mathbb{R}) The definition of complexification for a Lie group G is a complex Lie group G_C with a continuous homomorphism \phi: G\to G_C with the universal property that, if f: G → H is an arbitrary ... 1 vote 0 answers 40 views ### Commutative monoid "generating" algebraic number field Let's say that we have an algebraic number field, which is also an ordered field (e.g. no roots of -1), and whose ring of integers is a UFD. We would like to form a multiplicative, commutative free ... 1 vote 1 answer 105 views ### Dependence of strict transform on the subscheme along which we blow up It is stated in https://stacks.math.columbia.edu/tag/080C that Note that taking the strict transform along a blowup depends on the closed subscheme used for the blowup (and not just on the morphism ... 3 votes 1 answer 73 views ### Can a product map factor up to homotopy through the fat wedge For n \geq 2, for i = 1,\dots,n, let f_i \colon X_i \longrightarrow Y_i be maps of based CW-complexes, and consider a map$$f \colon \prod_{i=1}^n X_i \longrightarrow \prod_{i=1}^n Y_i such ...
Given $R$-module maps $\{f_i \colon M_i \to N_i\}_{i \in I}$, the universal properties of the direct sums $\oplus M_i$ and $\oplus N_i$ give a unique map $\oplus f_i \colon \oplus M_i \to \oplus N_i$ ...