Questions tagged [universal-property]
The universal-property tag has no usage guidance.
26
questions with no upvoted or accepted answers
9
votes
0
answers
122
views
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 ...
9
votes
0
answers
339
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}(...
8
votes
0
answers
94
views
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 ...
8
votes
0
answers
156
views
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 ...
8
votes
0
answers
143
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 ...
8
votes
0
answers
160
views
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(\...
8
votes
0
answers
392
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 ...
7
votes
0
answers
158
views
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 ...
7
votes
0
answers
361
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 ...
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^{\...
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 ...
4
votes
0
answers
332
views
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 ...
4
votes
0
answers
195
views
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$},\\
...
3
votes
0
answers
50
views
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$,...
3
votes
0
answers
102
views
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 ...
3
votes
0
answers
97
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. ...
2
votes
0
answers
114
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 ...
2
votes
0
answers
184
views
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$.
...
2
votes
0
answers
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$ ...
2
votes
0
answers
90
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 ...
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 ...
1
vote
0
answers
238
views
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 ...
1
vote
0
answers
55
views
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.
...
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 ~:~ \...
0
votes
0
answers
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, ...
0
votes
0
answers
63
views
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 ...