Questions tagged [yoneda-lemma]
The yoneda-lemma tag has no usage guidance.
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"Philosophical" meaning of the Yoneda Lemma
The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward.
Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
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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 ...
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In what generality does Eilenberg-Watts hold?
In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...
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The Yoneda Lemma for $(\infty,1)$-categories?
According to this page on the nLab, it is currently unclear whether or not the entire Yoneda lemma generalizes to $(\infty ,1)$-categories rather than just the Yoneda embedding. Have there been ...
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Subcategories which still give a Yoneda embedding
If $\mathbf{C}$ is a category, then the Yoneda functor which sends $a$ to $Hom_\mathbf{C}(-,a)$ is a fully faithful embedding of categories
$$ \mathbf{C}\rightarrow \mathbf{Func}(\mathbf{C}^{op},\...
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Yoneda Lemma for internal presheaves
I'm looking for a reference explaining under what conditions the internal Yoneda lemma holds; in particular, I am wondering if it is known what properties of the ($2$-)category of categories are ...
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Given a small category with some colimits, can the rest of the colimits be added?
Let $\mathcal{A}$ be a small category with some ( maybe no) colimits. What I would like to be able to do is add the rest of the colimits in a universal way. The Yoneda lemma will not work, since this ...
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In what context can enriched category theory be done?
There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.
My question is what ...
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Yoneda embedding and Horn sentences
The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.
Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, ...
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Yoneda on a not so small category
I am working with "usual" category theory, maybe over ZFC, and I have a functor $F : Set \to Set$. I'd like to apply Yoneda lemma to $F$, i.e. obtain:
$$ [Set, Set](h_A, F) \cong F A $$
However, ...
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Is $\prod_{X : \mathcal{U}} (X \to X) \cong 1$ consistent with type theory?
Assume we work in some minimalistic version of Martin-Löf type theory. Does it break consistency to postulate that the function that selects the identity function has an inverse?
$$\prod_{X : \...
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Enrichments vs Internal homs
Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor
$$
X \otimes -: \cal{C} \to \cal{C},
$$
for ...
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generalized elements in monoidal categories
In a category $\mathcal{C}$, a generalized element of an object $A$ means a morphism to $A$. It follows from Yoneda lemma that the object $A$ is determined by the collection of generalized of elements ...
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Has the "Lambek embedding" into the category of (co)product-preserving presheaves been studied very much?
The Lambek embedding is a particular embedding which is similar to the Yoneda embedding.
Suppose we have any category $C$. Recall that a presheaf on $C$ is defined as a contravariant functor from $C$ ...
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Yoneda lemma for monoidal categories
I am looking at the Yoneda lemma trying to see where the assumption of "locally small" really comes in. Obviously in order to define a functor to the category sets using $Hom$-spaces we need ...
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Unicity of Yoneda isomorphism
I am wondering if there is only one unique Yoneda isomorphism, that is a natural isomorphism (natural in C and P, that is) between Hom(yC,P) and PC.
The Yoneda lemma says that there exists at least ...
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Yoneda map for a composition of a representable functor and an arbitrary functor
Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So ...
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Higher and lower analogues of Yoneda's lemma
Here's a statement of Yoneda's lemma for n-category.
Let C be a n-category and $C^{\wedge}=[C^o,n-1Cat]$ be the n-category of presheaves on C.
$C^o$ is the opposite n-category of C and $n-1Cat$ is ...
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How to learn about Higher Topoi
I have been learning quasicategory theory in an attempt to understand higher topoi, and I have been trying to look at as many different sources as possible. Higher Topos Theory provides a higher ...
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Does Cantor Bernstein hold in a Closed Symmetric Monoidal Category?
In a closed symmetric monoidal category with $[I,X] \cong X$ for all $X$ is it true that having monomorphisms $m :A \rightarrow B$ and $m: B \rightarrow A$ is enough to imply $A \cong B$ ?
I tried to ...
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When do Kan extensions preserve colimits?
Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...
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Proof without using Yoneda's lemma?
Let $\mathscr{T}$ be atriangulated category.
The third axiom for triangulated categories, namely,
if in the diagram
$$\begin{array} 0X &\stackrel{u}{\longrightarrow}&Y&\stackrel{v}{\...
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Is Cauchy completion the largest extension with the same free cocompletion?
EDIT Title has been edited.
Let $C$ be a category, and $$\hat{C} = [C^{op}, (Set)]$$ be its free cocompletion. Despite its name, the free cocompletion of free cocompletion is not equivalent to the ...
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Why are functor categories nice? [migrated]
I was looking at the Yoneda embedding and one motivation is that we are embedding the category into a functor category and "functor categories are nice". What does this mean? What nice ...
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Yoneda Lemma from the perspective of "Categories = Partial Semigroups"
Categories can equivalently be defined as a special kind of partial semigroup: We impose some axioms on a partial semigroup that ensure the existence of well behaved "(partial) identity elements&...
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Arrows, furnished by Yoneda
What are some examples of 'important arrows' in a category that are significantly easier to define via fullness of the Yoneda embedding than in the base category?
The example that brought this to ...
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Profunctors as a Kleisli bicategory
There is some discussion on the nLab on seeing the free cocompletion $\mathbf{Psh}(\mathbf{A}) = [\mathbf{A}^{op}, \mathbf{Sets}]$, as a pseudomonad. The Yoneda embedding $よ \colon \mathbf{A} \to \...
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Does the following characterize local presentability?
Let $\mathcal C$ be a cocomplete category. Consider the following two conditions:
$\mathcal C$ is locally presentable.
The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors } ...
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Yoneda as a dinatural transformation 'up to iso'
$\newcommand{\op}{\mathrm{op}}$For a locally small category $\mathcal{C}$, let $y_\mathcal{C}:\mathcal{C}\to{\bf Set}^{\mathcal{C}^{\op}}$ denote the Yoneda embedding at $\mathcal{C}$. Letting ${\bf ...
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If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?
Let $T = (t, \mu, \eta)$ be a monad on an object $A$ of a 2-category $\mathcal K$. In The formal theory of monads, Street proves (Theorem 3) that if $l \dashv r$ is the canonical adjunction associated ...
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All functors "are" left adjoints, and applications?
Throughout this thread, let us assume smallness.
All functors "are" left adjoints
Let $D \xrightarrow{F} C$ be any functor, which induces
$$ D \xrightarrow{F} \hat{C}$$
by compositing the ...
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Are there any detailed references for the enriched yoneda lemma?
I am just starting out learning enriched category theory, and I am looking for a reference proof of the Yoneda lemma for categories enriched in a monoidal category.
Thank you for your help.
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Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary
Consider the word a.k.a. list monad $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$:
...
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Reference Request(Enriched Categories): Metric on Lipschitz Continuous Functions
If we consider metric spaces to be categories enriched over $\mathbb R_{\geq 0}$, the object corresponding to presheaves should be lipschitz-continuous functions $\operatorname{Lip^ 1}(M, \mathbb R_{\...
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Is the Kolmogorov-Arnold representation theorem an example of the Yoneda lemma?
From Wikipedia:
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold_representation_theorem
In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or ...
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Yoneda lemma for one object categories
Let $G$ be a group and let $\mathbb{G}$ be the associated one object category. Is there an explicit presentation of representable functors from $\mathbb{G} \to $Set? If so how does the Yoneda lemma ...
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Is there a bicategorical Yoneda lemma for marked lax transformations?
The bicategorical Yoneda lemma (see [Johnson–Yau, Chapter 8]) states that, given a bicategory $\mathcal{C}$ and a pseudopresheaf $\mathcal{F}\colon\mathcal{C}^{\mathsf{op}}\to\mathsf{Cats}_{\mathsf{2}}...
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Continuity of Kan extension along the Yoneda embedding
Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a ...
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Basic category theory: Universality of adjunction unit is justified by Yoneda Proposition in Mac Lane's text [closed]
Background
I am reviewing some category theory, which I did not learn too well the first time around. One text I am using is Mac Lane's. Near the beginning of the chapter on adjunctions (pg 80),
he ...
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