Questions tagged [universal-property]

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Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions: Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
hasManyStupidQuestions's user avatar
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117 views

Universal property of Isbell duality

Let's take $\mathrm{C}$ be a category, let's have an adjunction $(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves(C)} \leftrightarrows \mathrm{Presheaves(C)}$. One such adjunction is ...
Ilk's user avatar
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Adjoining a morphism to a finitely complete category

Let $\mathscr C$ be a finitely complete category. Let $x, y$ be objects of $\mathscr C$. We can describe the universal property of freely adjoining a morphism $x \to y$ to $\mathscr C$: it comprises a ...
varkor's user avatar
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Is totality a (large) cocompleteness condition?

A locally small category $A$ is called total if its Yoneda embedding $A \to [A^\circ, \mathbf{Set}]$ has a left adjoint. Such categories are necessarily small-cocomplete (since the presheaf category ...
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Is the universal object over a Hilbert scheme connected?

Hartshorne proved in his thesis that if $S$ is connected, then the Hilbert scheme $\operatorname{Hilb}^p=\operatorname{Hilb}^p(\mathbb{P}^n_S/S)$ is too (where $p\in \mathbb{Q}[z]$). Can the same be ...
Nathan Lowry's user avatar
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159 views

When does this commutative non-associative algebra have nilpotent elements?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dotsc, ...
mick's user avatar
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10 votes
1 answer
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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:\...
Madeleine Birchfield's user avatar
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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 ...
Branimir Ćaćić's user avatar
6 votes
0 answers
244 views

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^{\...
John Pardon's user avatar
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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 ...
Theo Johnson-Freyd's user avatar
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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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3 answers
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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 $R$},\\ ...
Oliver Nash's user avatar
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15 votes
2 answers
633 views

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 ...
varkor's user avatar
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8 votes
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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 ...
Johan Thiborg-Ericson's user avatar
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150 views

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 ...
varkor's user avatar
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7 votes
3 answers
446 views

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 ...
varkor's user avatar
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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 ...
Vanni's user avatar
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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(\...
varkor's user avatar
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11 votes
2 answers
945 views

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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2 votes
0 answers
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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 $T$. ...
Bargabbiati's user avatar
1 vote
1 answer
74 views

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 $N\times N$ ...
valle's user avatar
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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 ...
Ben MacAdam's user avatar
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6 votes
3 answers
412 views

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 $\...
Martin Brandenburg's user avatar
10 votes
3 answers
1k views

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 ...
Nadia SUSY's user avatar
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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. ...
Xiaosong Peng's user avatar
10 votes
1 answer
486 views

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 ...
geodude's user avatar
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1 vote
0 answers
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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 $k$...
ChaPi's user avatar
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1 answer
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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 ...
Alec Rhea's user avatar
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20 votes
2 answers
2k 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 ...
Luke's user avatar
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6 votes
0 answers
372 views

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 ...
aytio's user avatar
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0 answers
342 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}(...
Adrian Clough's user avatar
3 votes
0 answers
362 views

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 $U$ ...
a_g's user avatar
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9 votes
2 answers
2k 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, ...
Rorsa's user avatar
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2 votes
0 answers
91 views

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 ...
მამუკა ჯიბლაძე's user avatar
7 votes
1 answer
203 views

"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 ...
Robin Saunders's user avatar
4 votes
1 answer
1k 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 ...
Almeo Maus's user avatar
3 votes
0 answers
98 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. ...
Daron's user avatar
  • 1,761
0 votes
1 answer
292 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 ...
Manuel Bärenz's user avatar
11 votes
4 answers
2k 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 ...
Qiaochu Yuan's user avatar
5 votes
1 answer
417 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}...
Zhaoting Wei's user avatar
  • 8,707
0 votes
1 answer
567 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 ...
IMeasy's user avatar
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5 votes
1 answer
510 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 ...
Jason Gross's user avatar
6 votes
1 answer
835 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 $\...
The Prester's user avatar
12 votes
1 answer
2k 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 ...
Dmitry V's user avatar
  • 433
20 votes
5 answers
4k views

Universal property of the smash product (of pointed spaces)

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 in ...
Joseph Victor's user avatar
1 vote
0 answers
1k 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 ~:~ \...
Martin Brandenburg's user avatar
13 votes
2 answers
2k 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 ...
Martin Brandenburg's user avatar