13
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 Proposition 9.1 of André's Categories of Functors and Adjoint Functors (1966).
There is an earlier reference ...
- 10.6k
11
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 $\mathscr{C}$.
...
- 5,539
11
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 ...
- 66.7k
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 ...
- 27.7k
8
votes
Arrows, furnished by Yoneda
An example similar in spirit to yours is giving explicit examples of affine group schemes. Take, for example, $GL_n$: if we wanted to work solely in $\text{Aff} = \text{CRing}^{op}$ we'd have to write ...
8
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 ...
- 63.9k
8
votes
Accepted
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 ...
- 213
8
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).
7
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'...
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 ...
- 506
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 \...
- 25.2k
5
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 ...
- 63.9k
5
votes
Has the "Lambek embedding" into the category of (co)product-preserving presheaves been studied very much?
This construction is well known in enriched category theory, and has been studied in the increased generality of a class of weights (which recovers the finite discrete weights, describing finite (co)...
- 10.6k
4
votes
Has the "Lambek embedding" into the category of (co)product-preserving presheaves been studied very much?
It's not quite the same thing, but the full subcategory of presheaves preserving finite products (say call it $C^{f\times}$) is studied by Lurie in Higher Topos Theory section 5.5.8.
While $C\to\...
- 18.7k
4
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 ...
- 9,111
3
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}(...
- 467
3
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 ...
- 3,349
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 ...
- 9,111
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 ...
- 5,230
3
votes
Accepted
Yoneda as a dinatural transformation 'up to iso'
There is, in fact, a rather big picture studying this kind of phenomena "in general", i.e. not only for the Yoneda embedding into presheaves of sets:
R. Street, "Conspectus of variable ...
- 13.6k
2
votes
Accepted
Yoneda lemma for one object categories
Let me write $\def\B{\mathbf{B}}\B G$ for what you call $\mathbb{G}$.
You can check that presheaves on $\B G$ are precisely the right $G$-sets: the unique point of $\B G$ is sent to some set $X$, and ...
- 627
2
votes
Accepted
If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?
The answer is no: it is possible to have a terminal resolution without having an Eilenberg–Moore object.
Consider the 2-category $\mathbf{DagCat}$ of dagger categories, dagger functors, and natural ...
- 10.6k
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&...
- 63.9k
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
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 ...
- 4,459
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