12 votes
Accepted

Original reference for categories of presheaves as free cocompletions of small categories

The earliest reference I can find to the universal property of the presheaf construction is Remark 2.29 of Ulmer's Properties of Dense and Relative Adjoint Functors (1968). However, the proof is only ...
  • 5,520
10 votes
Accepted

Yoneda Lemma for internal presheaves

The Yoneda lemma holds in any finitely complete $2$-category $\mathscr{K}$, such as the $2$-category $\text{Cat}(\mathscr{C})$ of internal categories in a finitely complete category $\mathscr{C}$. ...
10 votes
Accepted

Yoneda lemma for monoidal categories

The Yoneda lemma is a purely formal result that does not require any size assumptions. For any closed symmetric monoidal category $\mathbf{V}$, any $\mathbf{V}$-category $C$, any object $A\in C$, and ...
  • 60.2k
9 votes

Original reference for categories of presheaves as free cocompletions of small categories

This isn't a direct answer to your question, but you might find the answer by looking in here: Joachim Lambek, Completions of Categories. Springer Lecture Notes in Mathematics 24, 1966. The ...
  • 25.8k
8 votes

Is $\prod_{X : \mathcal{U}} (X \to X) \cong 1$ consistent with type theory?

$\prod_{X : \mathcal{U}} (X \to X) \cong 1$ is consistent. It follows from parametricity and function extensionality. Usual parametric models also support function extensionality. The simplest one ...
7 votes
Accepted

Yoneda map for a composition of a representable functor and an arbitrary functor

This property of the functor $T$ is called being a dense functor, or "densely generating functor". The notion was first introduced by Isbell in the case where $T$ is fully faithful under the ...
  • 51k
6 votes

Higher and lower analogues of Yoneda's lemma

The fact that you have to consider setiods shouldn't be too surprising, since if we're working ($n>0$)-categorically the distinction between sets and setoids isn't invariant under equivalence. If ...
6 votes

The Yoneda Lemma for $(\infty,1)$-categories?

I realize it is a bit late to respond to the question... but this is precisely Lemma 5.1.5.2 of HTT (and the proof is very much along the same lines as the accepted one).
5 votes
Accepted

Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

This seems like too much to hope for at this level of generality. For example take $T$ to be the "(-1)-truncation" functor $$ T(\varnothing) = \varnothing, \qquad T(M) = * \,\, \mbox{for $M \...
5 votes
Accepted

Does the following characterize local presentability?

The following argument is from Martin Brandenburg's comment to the question linked to by Omar Antolin-Camarena. As you say, by the Yoneda lemma condition #2 holds iff every continuous functor $C^{op} \...
5 votes

"Philosophical" meaning of the Yoneda Lemma

This perspective seems to be absent so far, even though this is a very old question. To properly credit the source for this idea, the perspective seems to be taken for granted in MacLane and Moerdijk'...
4 votes
Accepted

Enrichments vs Internal homs

As Mike says, it is indeed the case that a closed monoidal structure gives rise to a self-enrichment of a category. This is an often-used fact. Here's an example of "wrong-way" self-enrichment: the ...
  • 51k
3 votes
Accepted

Profunctors as a Kleisli bicategory

This beautiful story take place in $\mathsf{Cat}$, the bicategory of locally small categories. There, the construction of small presheaves induces a pseudomonad whose pseudoalgebrabas are cocomplete ...
3 votes

Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

A similar notion (at least in spirit) was introduced by Simon Henry in An abstract elementary class non-axiomatizable in $L_{(\infty,\kappa)}$, see Def. 4.2 and later used by Michael Lieberman, Jiří ...
3 votes

Is Cauchy completion the largest extension with the same free cocompletion?

The answer is positive. I found a published account with details to be chapter 6 and 7 of Handbook of Categorical Algebra 1 by Francis Borceux. Thanks to the comments, useful links that summarize how ...
  • 4,207
2 votes

If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?

I don't know the answer to the original question, but I don't think I'd be alone in arguing that the condition "there is an adjunction terminal among those inducing $T$" doesn't "feel&...
  • 51k
2 votes

Profunctors as a Kleisli bicategory

A nice account of this is given in Martin Hyland's Elements of a theory of algebraic theories.
  • 121
2 votes

Enrichments vs Internal homs

I see that this question has an already accepted answer but I think that may be of interest. There is a notion of category with internal hom with no reference to a monoidal structure, that is the ...
2 votes
Accepted

generalized elements in monoidal categories

If I'm not wrong, the following is an answer to 2. For any objects $A,B,C$ of $\mathcal{C}$, we have the following natural map \begin{align} \Phi\colon\mathrm{Hom}(A\otimes B,C) &\to \mathrm{Nat}(...
  • 405
1 vote

Higher and lower analogues of Yoneda's lemma

We can try to apply the Yoneda lemma to $(n,k)$-categories for all $n$ and $k$. If we try to do this for $(0,0)$-categories, then we need the $(0,0)$-category of $(-1,-1)$-categories, and I'm not sure ...
  • 4,165

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