# Questions tagged [yoneda-lemma]

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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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### 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 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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### 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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### 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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### 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 ...