# Questions tagged [yoneda-lemma]

For questions related to Yoneda lemma in category theory, which *basically* says (among many other things it says) that every locally small category can be embedded nicely into it's functor category into the category of Sets. Use this tag along with (abstract-algebra) or (category-theory).

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### Why is el(-)=$\int(-)$ a functor from functors to a slice cateory?

I am reading up a little on category theory. I am trying to solve Riehl's problem (Category Theory in Context) 2.4.vii on page 72. I believe that it should be quite easy, but it appears to me that ...
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### Two questions on Yoneda's lemma

I used to be okay with the Yoneda lemma because in the material I studied it was only used in order to speed up a bit the discussion of universal objects that came up (tensor products or spectra for ...
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### How does the $2$-Yoneda embedding for the category of categories act on 2-morphisms?

Let $\text{Cat}$ be the category of small categories. I am interested in the Yoneda embedding $$Y : \text{Cat}^{op} \rightarrow [\text{Cat}, \text{Cat}]$$ $Y$ is a $2$-functor- it can be applied to ...
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### For which categories is its Yoneda embedding essentially surjective?

For a locally small category $\mathcal{C}$, you can embed $\mathcal{C}$ in the functor category $\mathrm{Set}^\mathcal{C}$ via the functor $X \mapsto \mathrm{Hom}_\mathcal{C}(X,{-})$. This embedding ...
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### Using Yoneda's Lemma to show that limits are unique

I recall my professor mentioning that Yoneda's Lemma can be used to show that limits of a functor are unique up to isomorphism. Here is my attempt: Let $F:J\to\mathcal{C}$ be a functor and let $X$ ...
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### Left and right Kan extensions

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between small categories. We define the functor \begin{align*} f:\hat{\mathcal{D}}&\longrightarrow\hat{\mathcal{C}} \\ G&\longmapsto G\circ F^{\...
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### Is the Yoneda completion of the rationals the extended real line?

A thing that I have often heard is that, when viewed as an enriched category, the Yoneda embedding of the poset of rationals $\mathbb{Q}$ into its category of presheaves is just the dense embedding of ...
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### Extension of a functor by colimits: Cisinski - Higher Categories and Homotopical Algebra - Remark 1.1.11

First, I state premilinary results. For a presheaf $X\colon A^{op}\to\mathsf{Set}$, it's category of elements, denoted by $\int X$, has pairs $(a,s)$ where $a \in A$ and $s \in X(a)$ as objects and ...
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### By Yoneda lemma, kernel of morphism in preadditive category is unique

I'm reading https://stacks.math.columbia.edu/tag/09SE. It says How do we use Yoneda lemma to prove the uniqueness of the kernel of a morphism? I tried this: Please help, thanks.
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### Cisinski, Higher Categories and Homotopical Algebra, Theorem 1.1.10

Let $X\colon \mathsf{C^{op}}\to \mathsf{Set}$ be a presheaf. It's category of elements, denoted by $\int X$, has pairs $(a,s)$ with $s \in X(a)$ as objects and $f \in \mathrm{Hom}_{\mathsf{D}}(a,b)$ ...
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### How does this statement reduces to the Yoneda lemma?

Context: I am reading the following page in the nLab, which is about simplicial presheaves, i.e Functors $\mathscr{C}^{\mathrm{op}} \to [\mathbf{\Delta}^{\mathrm{op}}, \mathbf{Set}]$. Equivalently, ...
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### Proving exponential arithmetic rules using the Yoneda lemma

Working in a Cartesian closed category we have have an exponential object $X^Y$ for object $X$ and $Y$. There are isomorphisms $1^X \cong 1$, $X^Y \times X^Z \cong X^{Y \times Z}$, $X^1 \cong X$ and a ...
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### Why is the Yoneda embedding continuous?

I realize this is a duplicate of this question, but I do not understand the answer. I am talking about the contravariant Yoneda embedding and this is how far I got: Let $D: I \to \mathcal{C}$ be a ...
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### Not getting example to fit Yoneda

So apparently I'm doing something wrong, but can't figure out what. I've read a proof of the Yoneda lemma and understand it from Riehl's book, but to try it out, I tried an example. My input category:...
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### Clarification in Yoneda Lemma exercise from Aluffi's Algebra Chapter $0$

I don't understand the sentence "Map $h_X$ to the image of $\mathrm{id}_X \in h_x(X)$ in $\mathscr F(X)$." $h_X$ is a functor, so, what can that sentence mean?