# Questions tagged [universal-property]

The universal-property tag has no usage guidance.

58
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### Universal property of the set of injections in the category of sets

Given two sets $A$ and $B$, the function set $B^A$ is characterized by the universal property that the functor $(-)^A:\mathrm{Set} \to \mathrm{Set}$ is the right adjoint of the functor $(-)\times A:\...

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### Constructing new categories by adding structure

On the one hand, let $\mathcal{C}$ be the monoidal category of finite-dimensional complex vector spaces and linear transformations, and on the other, let $\mathcal{D}$ be the monoidal category of ...

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### Universal property of dg-algebras

Let $k$ be a field. Does the fully faithful inclusion from $k$-algebras to dg-$k$-agebras concentrated in cohomological degrees $\leq 0$
$$\operatorname{Alg}_k\hookrightarrow\operatorname{dgAlg}_k^{\...

3
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0
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### Is there a name for objects all of whose endomorphisms are automorphisms?

I am looking for a descriptive adjective to describe the following special property that some objects in some categories enjoy: their endomorphism monoids are groups. Of course, one way this could ...

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### Universal property of the V-Mat construction

Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a ...

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### Proving the graded structure of the tensor algebra from only the universal property

When the tensor algebra is presented, it is usually constructed as the direct sum of all tensor powers of the space. By this construction, the graded structure of the tensor algebra is easy to prove. ...

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### Adjoining extensions in bicategories

Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that ...

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### Is the natural map from the free Lie algebra to the free associative algebra injective?

$\newcommand{\im}{\operatorname{im}}$Given a set $X$ and non-zero unital commutative ring $R$, let:
\begin{align}
A &= \mbox{free unital, associative algebra on $X$ with coefficients in $R$},\\
...

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2
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### Original reference for categories of presheaves as free cocompletions of small categories

It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A ...

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1
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### Does the morphism of composition have some universal property?

Let $A$, $B$ and $C$ be three objects in the category Set. For simplicity, assume that their underlying sets contain a finite number of elements, a, b and c respectively. Using the usual Haskell ...

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### Original reference for the Fam construction

For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given ...

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### Prof and the completion of Cat under right adjoints

In Bénabou's Les distributeurs, in which the bicategory of profunctors is introduced, Bénabou remarks (page 17, quoted below) that $\mathbf{Prof}$ may be viewed as the construction of a bicategory ...

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### Proof of uniqueness in the universal property of Poincaré line bundles

My question concerns the proof of a part of Lemma IV.2.2 (pag. 168) of the book Geometry of Algebraic Curves. vol. I by Arbarello, Cornalba, Griffiths and Harris. In order to state my problem, let me ...

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### What is the relationship between free bicompletion and the Isbell envelope?

Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\...

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### How to understand adjoint functors?

I asked this same question on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here.
I have a good ...

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### if $\max_{z \in K} |\zeta(z+it)-f(z)|<\epsilon.$ then is this $\lim_{t\to \infty} \inf \frac {|\zeta'(z+it)|}{|f'(z)|} $ a finit limit?

Universality theorem of Riemann zeta function states that :Let $K$ be a compact subset with connected complement lying in the strip $\{1/2 < \operatorname{Re}(z)<1\}$, and let $f : K \rightarrow ...

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### Group law in universal central extension of Thompson's group T

I'm having some troubles with universal central extensions and associated cocycles; in particular, I want to understand the group law of the universal central extension of the Thompson's group $T$.
...

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### Universality of the top eigenvalue of correlation matrices

Let $X$ be a $N\times P$ matrix with random independent and identically distributed entries $x_{ij}$. I also assume that $\langle x\rangle = 0$ and $\langle x^2\rangle = 1$.
Define the $N\times N$ ...

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### When will a monad satisfy Moggi’s “equalizer” property?

I’m interested in finding when ($V$)-monads will satisfy the universal property that $\eta$ equalizes $\eta_T, T\eta$. I’m particularly interested in the case for monads on presheaf categories or that ...

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### Universal property of the cocomplete category of models of a limit sketch

Let $\mathscr{S}$ be a limit sketch in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\...

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### Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...

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### Questions of the paper "PBW-pairs of varieties of linear algebras"

I am reading this paper "PBW-pairs of varieties of linear algebras", the link is here:https://www.tandfonline.com/doi/abs/10.1080/00927872.2012.720867.
At page 672, there is a definition of PBW-pair.
...

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### Intuition behind orthogonality in category theory, and origin of name

In category theory, two morphisms $e:A\to B$ and $m:C\to D$ are said to be orthogonal if for any $f:A\to C$ and $g:B\to D$ with $m\circ f=g\circ e$, there exists a unique morphism $d:B\to C$ such that ...

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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 $k$...

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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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### 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 ...

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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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### 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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### 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 $U$ ...

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### 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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### 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 ...

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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 ...

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### 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 ...

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### 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. ...

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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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### 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 ...

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### 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}...

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### 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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### 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 ...

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### 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 $\...

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### 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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### 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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### 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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### 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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### 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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### 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}) \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...