# Tagged Questions

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### Citations for the proof of universality of graph classes

In Automorphisms of graphs, Peter J. Cameron mentioned following classes of graphs which are universal structures. graphs of valency k for any fixed k > 2; bipartite graphs; strongly regular graphs; ...
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### Universal object

I'm trying to define the concept of universal object in category theory, but I failed to find a universal way to do it. Maybe someone can help ? THe idea is quite simple. Let $\mathcal{C}$ be a ...
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### Would this proof be considered true.Proving a property of a operation

Ok so the operation [x] is defined to be equal to the integer such that it is $\leq x$ From this definition it holds that : $$[x] \leq x$$ I need to prove that $$[x+n] = [x] + n$$ My proof ...
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### How do I get from the universal product of the tensor product to other definitions.

I was wondering how you can "derive" the common (or "classical") definition of the tensor product before the universal property was established (I think there is no need to repeat it here), i.e. a ...
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### Localization and the universal property

I've been trying to get to grips with the universal property and was looking at the localization of a commutative ring $A$ at some multiplicative set $S$, denoted $S^{-1}A$, to get an idea of what the ...
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### On Schauder basic systems in universal enveloped algebra of system of countable family of bounded selfadjoint operators

Let $A = C^*(1,T_1,T_2, ... | T_i^* = T_i, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of selfadjoint operators. I want to know as more as possible about that algebra,...
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### What does “4-universal hash function” mean?

I encountered the notion of 4-universal hash function and I cannot understand what exactly it means. This article https://en.wikipedia.org/wiki/Universal_hashing did not really help to clarify it. ...
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I am thinking about the universal property of products: Let $X$ and $Y$ be objects of a category $D$. The product of $X$ and $Y$ is an object $X \times Y$ together with two morphisms $\pi_1 : X \... 1answer 38 views ### Proving a set is open in a locally ringed space$(X,\mathscr O_X)$Let$(X, \mathscr O_X)$be a locally ringed space and let$A = \Gamma(X,\mathscr O_X)$be the global sections. For$f\in A$, define the "distinguished open base" as: $$D(f) = \{x\in X : \pi_x(f) \not\... 2answers 439 views ### Tensor product of monoids and arbitrary algebraic structures Question. Do you know a specific example which demonstrates that the tensor product of monoids (as defined below) is not associative? Let C be the category of algebraic structures of a fixed type,... 0answers 54 views ### Monomorphisms of monoids are stable under coproducts Let M,N,K be three monoids (or even groups, if you like) and let N \to K be an injective homomorphism. Then, the induced morphism M \sqcup N \to M \sqcup K is also injective. This is easy to ... 1answer 39 views ### Finite graphs forms Fraïssé Class with limit Rado/random graph I was reading Peter Cameron's explanation of Fraïssé's Theorem in his book Permutation Groups (Chapter 5). He states the Fraïssé class of finite graphs has limit being the countable random/Rado graph, ... 0answers 62 views ### Grothendieck's definition of a universal problem In EGA I, Grothendieck writes (I'm paraphrasing): Let \mathbf{K} be a category, (A_{\alpha})_{\alpha \in I}, (A_{\alpha \beta})_{(\alpha,\beta) \in I \times I} two families of objects of \... 1answer 130 views ### What is the class of topological spaces X such that the functors \times X:\mathbf{Top}\to\mathbf{Top} have right adjoints? For any topological space X, define a functor \times X:\mathbf{Top}\to\mathbf{Top} by Y\mapsto Y\times X (and acting on the hom-sets in the natural way). I know that if X is locally compact, ... 2answers 217 views ### Does the forgetful functor from \mathbf{TopGrp} to \mathbf{Top} admit a left adjoint? Let TopGrp be the category of topological groups (not necessarily T_0) and Top the category of topological spaces. Does the forgetful functor U:\mathbf{TopGrp}\to\mathbf{Top} admit a left adjoint?... 1answer 52 views ### universal properties of dependent types What is the universal property of dependent product / dependent sum? (I want to see a diagrams) They are must be different from usual ones, aren't they? (i'm trying to understand category theory ... 1answer 45 views ### \{\alpha_i : A_i \to C_i \} family of maps \implies \exists ! \alpha : \prod_i A_i \to \prod_i C_i such that \pi_i^C \alpha \to \alpha_i \pi_i^A Suppose that \prod_i A_i, \prod_i C_i exist in a category, and that there is a family of maps \{\alpha_i : A_i \to C_i\}. There exists a unique \alpha : \prod_i A_i \to \prod_i C_i such that \... 1answer 49 views ### proof that there is a unique isomorphism between two free modules I'm trying to figure out if there's any way to shorten the following proof of Corollary 7 in Dummit & Foote (p. 355) so that I don't have to write down the expressions of any specific elements ... 0answers 79 views ### Universal property of l^p-spaces The category \mathsf{Ban_1} of Banach spaces together with short linear maps (i.e. those of norm \leq 1) seems to have a natural construction which interpolates between coproduct and product: Let ... 1answer 46 views ### Arrow From an Initial Object Does Not Disturb Commutativity of the Diagram Suppose I have a commutative diagram D in the category of abelian groups. In D there appear abelian groups A, B, A\oplus B and some other abelian groups. Also, the natural inclusions i:A\to ... 0answers 41 views ### Universal property of free products. In my textbook, the Van der Waerden trick is used to prove that every word is equivalent to exactly one reduced word. For this, the author used the universal property of free products, so I tried to ... 2answers 102 views ### Observability of a System in State Space form as we all know, each system has infinite state space representations. My question is, whether the observability/controllability (and other common properties) are a properties of the SYSTYEM or ... 1answer 41 views ### L(\bigoplus_{i\in I} H_i)\cong \prod_{i\in I}L(H_i) as Banach spaces? Let \{H_i:i\in I\} be a system of complex Hilbert spaces, L(\bigoplus_{i\in I} H_i) the set of bounded linear maps T:\bigoplus_{i\in I} H_i\to \bigoplus_{i\in I} H_i, equipped with the operator ... 1answer 28 views ### Existence of unique homomorphism between free groups We know that a free group F(S) on a set S comes equipped with a function \alpha_S : S \to F(S). For any function \beta: S \to G, for some group G, there exists a unique homomorphism \phi : F(S) ... 1answer 30 views ### Associativity of extension of scalar (tensor product) I am trying to prove the following basic property$$(M \otimes_A B) \otimes_B C \cong M \otimes_A (B \otimes_B C) \cong M \otimes_A C$$where M is A-module, B is an A-algebra and C is B-... 1answer 40 views ### Can the choice of definition of morphisms for a slice category be justified categorically? An example of a slice category (\mathscr{C} \downarrow c) derived from some fixed object c of some base category \mathscr{C}, would be one whose objects correspond to the \mathscr{C}-morphisms ... 2answers 225 views ### Is there a concept of a “free Hilbert space on a set”? I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set X. Before ... 1answer 183 views ### Help with diagram chasing Given the diagram \require{AMScd} \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V\alpha VV \#@V\beta V V\# @VV\gamma V @. \\ 0 @>>> {A'} @>>{... 0answers 141 views ### Does zero-kernel imply monic in Abelian categories? I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ... 1answer 50 views ### Notions of free (and/or cofree) Hopf algebras? I don't know if this somewhat vague question has a precise answer, but I'd appreciate thoughts and references. Let k be a field. I am looking at Hopf algebras over k by which I mean an algebra ... 1answer 99 views ### Universal property of quotient group Let H\le G be a normal subgroup of G. Then we can think a natural projection map \pi:G \to G/H. Then this map has the following universal property: Let \phi:G \to G' be a homomorphism. If H ... 1answer 58 views ### What's so special about the grounded poset of cardinality 2? By a grounded poset, I mean a poset P with a bottom element 0_P. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it ... 1answer 57 views ### Functions in the definition of Universal Mapping Property of a free monoid In Awodey's Category Theory (http://www.amazon.com/dp/0199587361/?tag=stackoverfl08-20) p.19, what is |\bar{f}|? What is its relation to \bar{f}, which is in the existence statement? Is it ... 1answer 76 views ### Unique isomorphisms and universal properties Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ... 1answer 38 views ### What does “universal w.r.t. this property” mean? (kernel of a morphism in an additive category) In an additive category \mathcal{A} a kernel of a morphism f: B\to C is defined to be a map i : A \to B such that fi = 0 and that is universal with respect to this property. This is quoted ... 1answer 81 views ### Universal Property for isomorphic objects Suppose A,B are isomorphic objects of some category C. Suppose that A satisfies a given universal property does B satisfy the same? If not can you provide a counterexample for an elementary ... 1answer 79 views ### How to construct a tensor product of two preadditive categories in pure categorical fashion? Let \mathsf C and \mathsf D be two preadditive categories (by an preadditive category I mean a category together with compatible abelian group structure on every hom-set). I thought of ... 3answers 73 views ### Given a subset S, is there any universal construction (property) of L(S), the linear span of S? Given a subset S of a vector space V, is there any universal construction (property) of L(S), the linear span of S (like the way we can construct the fraction field of an integral domain)? 0answers 100 views ### What should this definition be? This is from Advanced Linear Algebra by S.Roman. Definition (on page 356). Referring to the figure below, let A be a set and let \mathcal S be a family of sets. Let$$ \mathcal F = \{g\colon A\... 1answer 233 views ###$F$entire with$\lim_{k \rightarrow \infty} F(z + N_{k}) = h(z)$for every$h$entire Universal Entire Functions. Prove that there exists an entire function with the following "universal" property: Given any entire function$h$, there is an increasing sequence$\{ N_{k}\}_{k=1}^{\...
I'm interested in the universal properties of the Cantor set. It is well-know that the Cantor set $2^\mathbb{N}$ is "universal" in the category of metrizable compact spaces, in the sense that every ...
### Proving that the free group on two generators is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$
I want to show that the free group on two elements $F(\{x,y\})$ is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$. The idea is to use the universal property of free groups to prove that \$F(\{...