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0
votes
1answer
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
3answers
378 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
1answer
290 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
1answer
179 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
0answers
152 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
1answer
248 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
1answer
365 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
5answers
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
0answers
356 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
2answers
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
11answers
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
3answers
447 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
3answers
761 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
0answers
234 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
0answers
217 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
4answers
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
3answers
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
3answers
312 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 ...
7
votes
1answer
346 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
3answers
976 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
3answers
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
2answers
897 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
4answers
1k views

Is there a universal property for Witt vectors?

Do the Witt vectors satisfy a universal property?
9
votes
4answers
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 ...