Questions tagged [universal-property]

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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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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
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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 ...
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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
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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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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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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 ...
Sasha's user avatar
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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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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 ...
András Salamon's user avatar
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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^{\...
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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 ...
aytio'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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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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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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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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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
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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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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
2 votes
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360 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 $U$ ...
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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 ...
მამუკა ჯიბლაძე's user avatar
1 vote
0 answers
125 views

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 ...
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 ...
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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
1 vote
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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 ~:~ \...
Martin Brandenburg'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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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 ...
zeraoulia rafik's user avatar