# Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

331 questions
Filter by
Sorted by
Tagged with
42 views

20 views

### Making a “definition” by way of a universal property

This is somewhat a question out writing styles. In a paper, I want to introduce and "define" an object $X$ by just stating the universal properties which is satisfies. Logically, this is not quite ...
59 views

### If $S\to T$ is a closed immersion, then $X\times_S Y\simeq X\times_T Y$.

Let $X,Y$ be $S$-schemes and $S\to T$ a closed immersion of schemes. Prove that we have a natural $T$-isomorphism $X\times_S Y\simeq X\times_T Y$. Let $f:X\to S$ and $g:Y\to S$ the structural ...
39 views

55 views

### Doubt on hypothesis of universal property of direct sum

I have seen that the following theorem, known as universal property of the direct sum, is always stated for direct sums of abelian groups: (Universal mapping property of external direct sum): let ...
30 views

### How can I prove that an different compactification to the Stone-Cech compactification doen't have it universal property?

Let $S(X)=\prod_{U \in \tau_{X}} \{ 0,1 \}_{U}$ with $(X,\tau_{X})$ $T_0$ topological space and $\{ 0,1 \}$ the Sierpinski's space. Cosiderer the map $e$ from $X$ to $S(X)$. $e:X\rightarrow S(X)$ ...
25 views

### If a presheaf $\mathcal{F}$ coincides locally with a sheaf $\mathcal{G}$, then $\mathcal{F}^{sh}\simeq \mathcal{G}$?

I'm trying to prove whether or not this is true: Let $\mathcal{F}$ be a presheaf and $\mathcal{G}$ a sheaf (of rings, let's say) on a space $X$. If $\{U_i\}_i$ is a basis for the topology of $X$ ...
81 views

### Is the standard definition of tensor product an effective way to introduce it?

I was taught the tensor product via its universal property: the only object that satisfies ... up to isomorphism. Later, I literally discovered one could actually write down the elements of (some) ...
121 views

### Homomorphisms from $\prod_{i\in\mathbb Z}\mathbb Z$ to $\oplus_{i\in\mathbb Z}\mathbb Z$ that fixes $\oplus_{i\in\mathbb Z}\mathbb Z$

I'm trying to verify that $\prod_{i\in\mathbb Z}\mathbb Z$(the direct product of countably many $\mathbb Z$) is not a coproduct in the category of abelian groups. We know that the coproduct object is ...
16 views

88 views

100 views

### Question on coproducts and products

Given a family of modules $\{A_i\}_{i \in I}$, I always understood that the main difference between an element of the product $\Pi A_i$ and the direct sum $\oplus A_i$ to be that if you take an ...
26 views

### There is a unique map $P \to \text{Hom}(A, \prod_i C_i)$ whenever $P \xrightarrow{p_i} \text{Hom}(A, C_i)$?

I'm trying to prove directly that $\text{Hom}(A, \prod_i C_i) \simeq \prod_i \text{Hom}(A, C_i)$ whenever $\prod_{i\in I} C_i$ is a product of a family of objects $C_i$ in a category $B$, where $A$ is ...
90 views

### Kernel of group epimorphism with finitely presented is the normal closure of a finite subset .

Suppose $F : G \rightarrow K$ is a group epimorphism with $G$ finitely generated and $K$ finitely presented. Then, show that $H := \ker F$ is the normal closure of a finite subset in $H$. My attempt ...
188 views

### How does Universal Mapping Property encode “no-junk” and “no-noise” in free monoid?

I am going trough the "Category Theory" book by Steve Awodey. In the "1.7 Free categories" chapter the author introduces the following algebraic definition of free monoid: A ...
34 views

### Is the image a universal object?

Given a function $f:X\to Y$ in category $\mathcal{C}$, one can construct the image as a factorisation $f=(e:I\hookrightarrow Y)\circ(g:X\to I)$ that is universal (initial) among all such ...
30 views

### Product projections are a universal arrow?

From Mac Lane's Categories for the Working Mathematician: Let $S: D \rightarrow C$. A universal arrow arrow from $S$ to $c$ is a pair $\langle r,v \rangle$ consisting of an object $r \in D$ and an ...
81 views

### Cokernel within the Category of Groups

This question is in regards to Aluffi's Algebra: Chapter $0$, II.$8.22$ $\textbf{8.22: }$Let $\varphi: G \rightarrow G'$ be a group homomorphism, and let $N$ be the smallest normal subgroup ...