Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. Reference: Wikipedia.

The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.

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What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the tedious details concerning duality. Suppose that $U:…
Hans-Peter Stricker
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Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global…
DBr
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Does the box topology have a universal property?

Given a set of topological spaces $\{X_\alpha\}$, there are two main topologies we can give to the Cartesian product $\Pi_\alpha X_\alpha$: the product topology and the box topology. The product topology has the following universal property: given…
Keshav Srinivasan
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Universal property of de Rham differential.

Suppose $A$ is a commutative algebra over a field $k$. It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the module of Kahler differentials. By definition, it…
Sasha Patotski
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universal property in quotient topology

The following is a theorem in topology: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/\sim$ be the canonical projection. If $g : X → Z$ is a continuous map such that $a \sim b$ implies $g(a) = g(b)$ for…
user9464
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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? To be concrete, given an arbitrary topological…
Censi LI
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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 I explain my attempt for a definition of this,…
Corsin Pfister
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Show a free group has no relations directly from the universal property

The free group is often defined by its universal property. A group $F$ is said to be free on a subset $S$ with inclusion map $\iota : S \rightarrow F$ if for every group $G$ and set map $\phi:S \rightarrow G$ there exists a unique homomorphism…
cede
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Universal property of $N\rtimes K$

Given groups $N$ and $K$, if $K$ acts on $N$ by \begin{equation} K\xrightarrow{\theta}\operatorname{Aut_{Grp}}(N), \end{equation} then we can define a group $N\rtimes_{\theta}K$ whose elements are like in $N\times K$ but the multiplication is…
Hui Yu
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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, and let us denote by $|~|$ the underlying functor…
Martin Brandenburg
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What is a Universal Construction in Category Theory?

From pg. 59 of Categories for the Working Mathematician: Show that the construction of the polynomial ring $K[x]$ in an indeterminate $x$ over a commutative ring $K$ is a universal construction. Question: What does the author mean by this bolded…
user1770201
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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 $F(\{x,y\})$ satisfies the universal property of…
MAM
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Do Wikipedia, nLab and several books give a wrong definition of categorical limits?

It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and write some errata emails. A standard definition of a…
joriki
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How are universal properties “solutions to optimization problems”?

I have on multiple occasions stumbled on the idea that a universal property in category theory is a “most efficient solution to a problem”. See e.g. the wikipedia page. However, I don’t find the intuition given there very clarifying. I find this…
user56834
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Why do exponential objects (in category theory) require currying?

I'm a bit confused about exponential objects in category theory. I find them pretty intuitive when I think about them as "arbitrary-arity cartesian products"; in the sense that if I had never seen exponentials and you reminded me of the standard…
Nate
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