Questions tagged [universal-property]
For questions about universal properties of various mathematical constructions.
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Universal property definition of an ideal generated by a subset?
I'm puzzled by the definition of ideals generated by a subset of a ring in Aluffi, Algebra: Ch 0.
The previous chapter on groups is (for an algebra book of this level) quite categorical in spirit. In ...
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Universal property of HNN extensions
The amalgamated free product of groups can be defined as a certain presentation; or, one can define it as a pushout where the morphisms are embeddings. I was wondering if the definition of an HNN ...
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Equivalent definitions of tensor power of a vector space
I have two definitions of the tensor power $T^nV$ of a vector space $V\in \bf{kVect}$.
For every $n$-multilinear map $f:V\times ...\times V\rightarrow W$, there exists a unique $\bar{f}:T^nV\...
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Definite description in homotopy type theory
I asked this question there and I have been suggested to ask it here.
In a paper by David Corfield, we have an account of definite description in homotopy type theory. The author gives the following ...
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why is the universal side divisor called universal?
With respect to the usage of the term "universal" in category theory I struggle to see the connection? In terms of elements, if we were to let divisibility represent the morphism, then ...
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Confused about question on universal property inverse limit
Context:
Let $S = \varprojlim S_{i}$ be a projective limit and define for each $j \in I$ the projection map $f_{j} : S \rightarrow S_{j}$ by $f_{j}((x_{i})_{i \in I}) = x_{j}$.
Now, I have already ...
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Continuous function from a product topological space to a sum topological space
I denote by $\mathbf{2}$ the set $\{0, 1\}$ that is given the discrete topology. Let $X$ be some topological space. I denote by $\mathbf{2} \times X$ the topological space with the product topology ...
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Universal property of product
I'm trying to reconcile the commutative diagram for the universal property of the product with my understanding through a concrete example. I'm very confused by all of this, and would appreciate if ...
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Can we construct the exterior algebra just from simple multivectors?
$
\newcommand\K{\mathbb K}
\newcommand\Ext{\mathop{\textstyle\bigwedge}}
\newcommand\Lip{\mathrm{Lip}}
\newcommand\ev{\mathrm{ev}}
\newcommand\Gr{\mathrm{Gr}}
$Let $V$ be a finite-dimensional $\K$-...
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The pushout $F_\alpha(H)$ of $H \overset{\pi}{\longleftarrow} \mathbb{Z}^{\ast H} \overset{\alpha^{\ast H}}{\longrightarrow} G^{\ast H}$
$\require{AMScd}$
Definition 1:
If $G$ is a group and $X$ is a set, define $G^{\ast X} = \ast_{x \in X} G$. This is functorial in both $G$ and $X$:
If $G$ and $H$ are groups, $\varphi: G \to H$ is a ...
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Understanding the Product as a Universal Arrow from the Diagonal Functor
This is my first question and it has been asked before, but I'm not comfortable with the answer so I thought I'd raise it again.
In Mac Lane's Categories for the Working Mathematician it is claimed ...
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Serge Lang's Definition of a Free Group
Serge Lang says the following in his "Algebra":
We now consider the category $\mathfrak{C}$ whose objects are the maps of $S$ into groups. If $f:S\rightarrow G$ and $f':S\rightarrow G'$ are ...
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Why is this localization thin?
I am studying this paper by Malkiewich and Ponto. I am unsure about one claim.
Let $\Delta$ be the augmented simplex category. Denote by $\mathfrak{J}$ the wide subcategory of $\Delta$ consisting of ...
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Why the associated graded universal enveloping algebra of a Lie algebra is Poisson?
Let $L$ be a Lie algebra and $U(L)$ be its universal enveloping algebra. How the associated graded algebra $grU(L)$ of the universal enveloping algebra $U(L)$ can be a Poisson algebra? How the Poisson ...
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If A is idempotent then A+AB−ABA isidempotent for any square matrix B with the same dimension as A. [closed]
If A is idempotent then A+AB−ABA is idempotent for any square matrix B with the same dimension as A.
I have this question to solve and I tried squaring the entire expression and then simplifying it ...
