The universal-property tag has no wiki summary.

**0**

votes

**1**answer

108 views

### What is the universal property of being the maximal common subobject of two objects in a semisimple category?

Imagine a semisimple abelian category $\mathcal{C}$, for example representations of a finite group.
Take two (nonsimple) objects $X, Y$ that are subobjects of a common object $Z$, and decompose them ...

**4**

votes

**3**answers

405 views

### Where does Segal's category come from?

Segal's category $\Gamma$ is the skeleton of the category $\text{FinSet}_{\ast}$ of pointed finite sets. It is used to write down $\Gamma$-spaces, which are functors $\Gamma \to \text{Top}$ satisfying ...

**4**

votes

**1**answer

297 views

### Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping algebra?

Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as ...

**0**

votes

**1**answer

183 views

### universal families and maps to quotient stacks

Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductive group. Roughly ...

**3**

votes

**0**answers

157 views

### Do generalizations of adjoint functors, such as adjunctions in 2-categories, and multivariable adjunctions, have formulations in terms of something like universal morphisms?

I recently learned about two-variable adjunctions and multi-variable adjunctions, and about adjunctions in 2-categories. The nCatLab page on two-variable adjunctions talks about how to go from the ...

**5**

votes

**1**answer

260 views

### universal property of blow up for stacks?

I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is to be lifted to ...

**5**

votes

**1**answer

383 views

### Adjoint Functors as Initial Objects of Some Category

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...

**8**

votes

**5**answers

1k views

### Universal Property of the Smash Product (of pointed spaces)

Hey
Is there a universal property for the smash product (of pointed spaces or pointed CW-complexes or something of that ilk)? I've seen the smash product of spectra defined with a universal property ...

**1**

vote

**0**answers

363 views

### When do coproducts commute with filtered projective limits?

Let $C$ be a category with filtered projective limits and coproducts (denoted by $+$). Then for all filtered projective systems $\{X_i\}$, $\{Y_i\}$ in $C$ there is a canonical morphism
$$\alpha ~:~ ...

**10**

votes

**2**answers

1k views

### Elements in a localization - category theoretic approach

This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the ...

**21**

votes

**11**answers

1k views

### Characterizing specific “concrete” mathematical objects by abstract general properties

In this note by Tom Leinster the Banach space $\mathrm{L}^1[0,1]$ is recovered by "abstract nonsense" as the initial object of a certain category of (decorated) Banach spaces. So a function space, ...

**3**

votes

**3**answers

452 views

### Are schemes pushouts of neighbourhoods and formal neighbourhoods?

Hello,
I have two questions, the first less important.
Let $X$ be a scheme, $x \in X$ a schematic point.
What is an elegant way of defining/characterizing the map $\operatorname{Spec}(O_{X,x}) ...

**4**

votes

**3**answers

792 views

### universal property of stein factorization

This question has not the ambition of being very precise, instead it is more philosophical.
The question is the following: does the Stein factorization of a morphism have some kind of universal ...

**3**

votes

**0**answers

238 views

### Stable distributions for Lindeberg exchange strategy?

Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...

**5**

votes

**0**answers

222 views

### Universal nondegenrate map over the space of complete linear maps

Let $E$ and $F$ be vector spaces and assume for simplicity that $\dim E = \dim F = n$.
Recall that the moduli space of complete linear maps from $E$ to $F$ is the space $M(E,F)$ parameterizing all ...

**9**

votes

**4**answers

1k views

### Completion of the rationals to the reals as an inverse limit construction?

There is of course the standard construction of the reals by considering the set of sequences that are Cauchy with respect to the standard metric and taking the quotient by sequences that converge to ...

**17**

votes

**3**answers

1k views

### Is there a “categorical” description of Grothendieck's algebra of differential operators?

First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due ...

**4**

votes

**3**answers

323 views

### Characterizing nilpotents in a ring by a universal property

This is again a question asked to me by this user. He apparently quit using MO due to a busy time in personal and professional life and resulting difficulties in spending time here with patience. I am ...

**8**

votes

**1**answer

351 views

### Can minimal surfaces be characterized by some universal property?

As objects which are minimal (in some respect), this seems entirely plausible, but I'm not sure what category we should be working in, and what restrictions we would need, to actually have a situation ...

**0**

votes

**3**answers

983 views

### equality of elements in localization via universal property [unsolved!]

I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. a very nice example for this ...

**9**

votes

**3**answers

1k views

### “Albanese” schemes: When does an “initial abelian scheme” exist under a given scheme?

For a variety V, its Albenese variety Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety. Can we say something similar for an arbitrary ...

**12**

votes

**2**answers

920 views

### What is the universal property of associated graded?

Given a filtered vector space (or module over a ring) 0=V0⊆V1⊆...⊆V, you can construct the associated graded vector space gr(V)=⊕iVi+1/Vi. Does gr(V) satisfy a universal property? ...

**12**

votes

**4**answers

1k views

**9**

votes

**4**answers

2k views

### What is the universal property of normalization?

What is the universal property of normalization? I'm looking for an answer something like
If X is a scheme and Y→X is its
normalization, then the morphism
Y→X has property P and any ...