Questions tagged [yoneda-lemma]

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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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1answer
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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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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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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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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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165 views

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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1answer
202 views

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 ...
5
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1answer
290 views

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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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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1answer
317 views

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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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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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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361 views

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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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 ...
7
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1answer
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generalized elements in monodial 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 ...
5
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2answers
337 views

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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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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183 views

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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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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1answer
172 views

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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2answers
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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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1answer
435 views

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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0answers
286 views

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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1answer
926 views

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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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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1answer
523 views

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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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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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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1answer
171 views

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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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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“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 ...