Questions tagged [universal-property]
The universal-property tag has no usage guidance.
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Module of Kahler differentials for manifolds [duplicate]
Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...
2
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1answer
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Definition of $\in_c$ for power objects
On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...
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1answer
529 views
What is the correct definition of localisation of a category?
Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange.
There appears to be a discrepancy in the literature regarding the ...
5
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0answers
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closed substack of quotient stack
The question concerns quotient stacks. I am not very comfortable with stacks, so feel free to edit the question if I am saying nonsense. It is also the reason why I may be spelling in too much detail ...
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0answers
159 views
The category of elements corresponding to a coend as a higher colimit
Let $D: \mathcal{C} \to \mathbf{Set}$ be a diagram of sets, then we can obtain the colimit of $D$ as the set of connected components of the category of elements of $F$, which we denote by $\mathrm{el}(...
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0answers
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Categorical quotients for quasi-affine varieties
Let $X$ be an affine variety and let $G$ be a reductive algebraic group acting on $X$. Let $U \subset X$ be a $G$-invariant open set.
Under what hypothesis there exists a categorical quotient of $U$ ...
7
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2answers
589 views
Is there a useful limit or co-limit of a diagram that has only a single object?
I'm starting to study category theory kind of informally and everytime I read about the definitions of limits and co-limits, the first three examples are always the same:
terminal/initial objects,
...
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0answers
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Analogs of universal coverings for vector bundles
Not sure if I should use the soft-question tag here. Decided to wait with it :D
The universal covering of a (nice enough) space $X$ is a universal, in a suitable sense, representation of $X$ in the ...
6
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1answer
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“Universal embedding structures” in a general setting?
There are a number of famous results to the effect that "countable structures" of a certain type have a universal, "homogeneous" structure of the same type into which they all embed with various nice ...
4
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1answer
459 views
Left adjoint of pullback
In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states:
Indeed, the UMP of pullbacks essentially states that composition along
any function α is left adjoint to pullback ...
3
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0answers
93 views
How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?
For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
0
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1answer
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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 ...
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4answers
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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 ...
5
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1answer
327 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 $\text{Mat}...
0
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1answer
298 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 ...
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0answers
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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 ...
6
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1answer
465 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 $\...
10
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1answer
927 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 ...
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5answers
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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 ...
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0answers
682 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 ~:~ \...
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2answers
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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 ...
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14answers
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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, ...
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3answers
528 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
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3answers
1k 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 ...
5
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0answers
290 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 ...
7
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0answers
304 views
Universal non-degenerate 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 ...
10
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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 ...
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3answers
2k 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
356 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
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1answer
389 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 ...
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3answers
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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 ...
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3answers
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“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 ...
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4answers
2k views
What is the universal property of associated graded?
Given a filtered vector space (or module over a ring) $0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V$, you can construct the associated graded vector space $\mathrm{gr}\left(V\right)=\oplus_{i}V_{i+...
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4answers
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6answers
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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 ...
