# 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” $\mathcal{D}_1$,...
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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 ...
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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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### 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 ...
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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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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^{\... • 18.3k 4 votes 0 answers 335 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 ... • 53.1k 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 ... • 8,755 2 votes 3 answers 417 views ### 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. ... • 121 5 votes 1 answer 130 views ### 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 ... • 8,755 4 votes 0 answers 196 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},\\ ... • 1,404 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 ... • 8,755 8 votes 1 answer 450 views ### 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 ... 8 votes 0 answers 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 ... • 8,755 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 ... • 8,755 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 ... • 55 8 votes 0 answers 165 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(\... • 8,755 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 ... • 1,019 0 votes 0 answers 64 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 ... • 2,236 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. ... • 223 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 ... • 864 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 ... • 1,253 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 \... • 61.5k 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 ... • 815 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. ... 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 ... • 2,129 1 vote 0 answers 120 views ### 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... • 111 4 votes 1 answer 541 views ### 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 ... • 9,009 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 ... • 423 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 ... • 371 9 votes 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}(... 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 ... • 31 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, ... • 873 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 ... • 17.5k 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 ... • 3,522 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 ... • 507 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. ... • 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 ... • 5,545 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 ... • 115k 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}... • 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 ... • 3,717 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 ... • 695 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 \... 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 ... • 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 ... • 1,147 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 ~:~ \...
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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 ...