# 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).

124 questions
Filter by
Sorted by
Tagged with
35 views

### Quotient of a free group on a set

Suppose I have a free group $F_S$ on a set $S,$ with a nonempty proper subset $T.$ If $N$ is the smallest normal subgroup of $F_S$ containing $T$, I would like to show that $F_S/N$ is also free. There'...
46 views

### When left Kan extension preserve colimits

I am currently working with Kan extensions, and I have found a neat little fact, and I'd like to know if there is some literatture on it, and maybe some names and characterization I could use. ...
97 views

### Deducing Cayley's theorem from the Yoneda lemma using Wikipedia's recipe [duplicate]

I'm following Wikipedia in trying to prove that Cayley's theorem emerges as a particular case of the Yoneda lemma. In case that article gets edited, here's the screenshot: A couple of aspects in the ...
33 views

### About Leinster's discussion of Corollary 4.3.10

Corollary 4.3.10 in Leinster's text says that $H_A\cong H_{A'}\iff A\cong A'\iff H^A\cong H^{A'}$. He writes: "the corollary tells us that two objects are the same if and only if they look the ...
58 views

91 views

75 views

97 views

### 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 ...
134 views

### 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 $X$ ...
147 views

### 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^{\...
179 views

### 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 ...
61 views

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 $... 1answer 40 views ### By Yoneda lemma, kernel of morphism in preadditive category is unique I'm reading https://stacks.math.columbia.edu/tag/09SE. It says How do we use Yoneda lemma to prove the uniqueness of the kernel of a morphism? I tried this: Please help, thanks. 1answer 121 views ### Cisinski, Higher Categories and Homotopical Algebra, Theorem 1.1.10 Let$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)$... 1answer 122 views ### How does this statement reduces to the Yoneda lemma? Context: I am reading the following page in the nLab, which is about simplicial presheaves, i.e Functors$\mathscr{C}^{\mathrm{op}} \to [\mathbf{\Delta}^{\mathrm{op}}, \mathbf{Set}]$. Equivalently, ... 1answer 107 views ### Proving exponential arithmetic rules using the Yoneda lemma Working in a Cartesian closed category we have have an exponential object$X^Y$for object$X$and$Y$. There are isomorphisms$1^X \cong 1$,$X^Y \times X^Z \cong X^{Y + Z}$,$X^1 \cong X$and a few ... 1answer 120 views ### 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 ... 1answer 92 views ### 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:... 0answers 63 views ### Apply the Lemma of Yoneda Let$\mathcal{A}\to\textrm{Sets}$be a covariant functor,$X, X'\in\operatorname{Obj}(\mathcal{A})$objects, and$(X, \Phi), (X', \Phi')$be representations of$F$, i.e.$\Phi: F\to\operatorname{Hom}_{...
Emily Riehl's "Category Theory in Context", ${\rm Exercise}~2.2.{\rm ii}.$ Explain why the Yoneda lemma does not dualize to classify natural transformations from an arbitrary set-valued functor to ...