22
votes
Monoidal categories whose tensor has a left adjoint
If $\otimes : V \times V \to V$ has a left adjoint and $V$ has finite products then $\otimes$ preserves them in the sense that the natural map
$$(X \times Y) \otimes (Z \times W) \to (X \otimes Z) \...
- 119k
20
votes
Accepted
Surmounting set-theoretical difficulties in algebraic geometry
Let me start by discussing a bit the option of having a large class of generators. You might be interested in the notion of locally class-presentable.
To be precise here, I need to be a bit set-...
- 9,405
20
votes
Accepted
Monoidal categories whose tensor has a left adjoint
Just to clean up the $\epsilon$ of room left after Qiaochu's answer -- we can get rid of the extra hypotheses. I'll write $I$ for the monoidal unit and $1$ for the terminal object.
Assume that $(\ell,...
- 64.2k
19
votes
Accepted
Notation for "the" left adjoint functor
In EGA 0.1.5.2-3 (from the 1971 Springer edition) the right adjoint and the left adjoint of a functor $F$ are denoted by $F^{\rm ad}$ and ${}^{\rm ad}\!F$, respectively.
- 6,710
19
votes
Reference request: Who first proved that right adjoints preserve limits?
Daniel M. Kan defined adjoint functors in his paper Adjoint functors (written in 1956).
In Chapter II he defines limits and colimits of arbitrary small diagrams
and proves that the limit and colimit ...
- 38k
17
votes
Accepted
Minimal set of assumptions for set theory in order to do basic category theory
To complement Tom Leinster's answer, let me try to be specific:
To form the product category $\mathcal{C} \times \mathcal{D}$, we need ordered pairs, which we can get from the axiom of unordered ...
- 48.9k
16
votes
Accepted
Is this space the Stone–Čech compactification?
No, the closure of the image of $f$ in $Y$ is never the Stone-Čech compactification of $X$ unless $X$ is empty. In particular, consider the element $a\in Y$ which is $1$ on every coordinate. Note ...
- 31.2k
15
votes
Accepted
Can the category of S-local objects be reflective but not a localization by S?
Not in general, no - there must be some additional conditions on $S$, such as a saturation condition.
Consider for instance the presentable case. Then if $S$ is small, $Loc(S) $ is always reflexive, ...
- 16.6k
14
votes
Accepted
Upgrade adjunction to equivalence
Let $\mathcal C$ and $\mathcal D$ be two categories, and let $F\colon\mathcal C\longrightarrow \mathcal D$ and $G\colon\mathcal D\longrightarrow\mathcal C$ be two functors, with $F$ left adjoint to $G$...
- 15.6k
13
votes
Accepted
Explicit expression of the unstraightening functor
Wow, I have always thought that unstraightening has to be easier than straightening, but I've never actually looked at Lurie's treatment before, so I'm surprised to realize he defines straightening ...
- 64.2k
13
votes
Upgrade adjunction to equivalence
Another and probably more natural interpretation of the sentence in the Wikipedia article may be called "localizing an adjunction to an equivalence".
Let $\mathcal C$ and $\mathcal D$ be two ...
- 15.6k
13
votes
Minimal set of assumptions for set theory in order to do basic category theory
You ask what assumptions on sets are needed in a "normal first course on category theory". There are several possible kinds of answer, and this is an answer of the practical kind, i.e. from ...
- 27.8k
12
votes
A specific property of bi-adjunction
(This is not an answer to your question, just a long comment.)
There is something to be careful about with ambidextrous adjunctions. When we work with an ordinary adjunction we can rest assured that ...
- 119k
11
votes
Accepted
Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie
Let $\mathcal{C},\mathcal{D}$ two $\infty$-categories and $f:\mathcal{C}\to\mathcal{D}$ and $g:\mathcal{D}\to \mathcal{C}$ two functors.
Recall (HTT.5.2.2.7) that a natural transformation $u:1_{\...
- 16.5k
11
votes
Accepted
Characterization of functors whose right adjoint is monadic?
Let $F: C \to D$ be a left adjoint functor. I hope I'm not saying anything stupid, but I think you can just rephrase the two conditions of Beck Monadicity theorem in terms of the left adjoint:
The ...
- 42.6k
11
votes
Prof and the completion of Cat under right adjoints
The idea that passing from $Cat$ to $Prof$ is a great way to give functors adjoints features prominently in Richard Wood's theory of proarrow equipments. Rosebrugh and Wood showed in Proarrows and ...
- 64.2k
10
votes
Accepted
If a right adjoint to the product functor exists, must it be the diagonal?
For the first question: Yes.
Let $C$ be a category with finite products and let $\Delta=(\Delta_1,\Delta_2):C\to C\times C$ s.t. $$\times\dashv \Delta.$$
More specifically we find $$[X\times Y, Z]\...
- 3,259
10
votes
Which direction of the adjoint functor theorem is most useful?
In 1-category theory, the easy direction (a) is invoked all the time. The hard direction (b) doesn't have to be formally invoked very often, because most adjoints can be constructed by hand (and even ...
- 64.2k
9
votes
If a right adjoint to the product functor exists, must it be the diagonal?
Building on Garlef Wegart's work, the answer is yes in general. The idea is to show that if $C$ has a right adjoint to $\times$, then it is enriched in pointed sets, so that $C$ embeds fully ...
9
votes
Categories which are both monadic and comonadic over another category
I understand it like this: if a monad $M: C \to C$ has a right adjoint $K: C \to C$, then that right adjoint carries a comonad structure which is mated to the monad structure, and the category of $M$-...
- 53.4k
9
votes
Upgrade adjunction to equivalence
I think the author of the wikipedia article probably had in mind Leonid Positselski's first answer, where one restricts to the full subcategory of fixed points of the adjunction. Beware there is no ...
- 64.2k
9
votes
Accepted
$\infty$-natural transformations and adjunctions
1- Your natural transformation can be seen as a functor $C \to D^{\Delta^1}$, which therefore induces a commutative square of $\infty$-categories
$\require{AMScd} \begin{CD} C_{x/} @>>> D^{\...
- 16.6k
9
votes
Which functors preserve the number of connected components?
You don't give your definition of $\pi_0$ on $\text{Top}$, but since you mention left adjoints I assume it is the left adjoint of the inclusion $\text{Set} \to \text{Top}$ of discrete spaces into $\...
- 119k
9
votes
Accepted
Which spectra have a universal connective quotient?
This answer is about the $\infty$-categorical variant. This is a fancy way to say: on spaces of maps, the natural map
$$
Map(T',A) \to Map(T,A)
$$
is an equivalence for any connective $A$. Note that ...
- 52.9k
8
votes
Can adjoint linear transformations be naturally realized as adjoint functors?
Not exactly an answer to the question as posed, but it's worth noting that adjoint linear maps and adjoint functors can both be realized as instances of the same thing, namely morphisms in a Chu ...
- 67.1k
8
votes
Accepted
Adjoints of scalar extension and scalar coextension
If $X$ is an $R$-module, there is a natural map $M\otimes_RX\to\text{Hom}_R\left(\text{Hom}_R(X,R),M\right)$ given by $m\otimes x\mapsto[\varphi\mapsto m\varphi(x)]$ that is easily checked to be an ...
- 35.4k
8
votes
Adjunctions between Groupoids and Hilbert spaces
I don't know what morphisms you intend for the category of finite-dimensional Hilbert spaces, but it doesn't actually matter. The answer is no, there are no interesting adjunctions between the ...
- 64.2k
8
votes
Adjoints for radical and socle functors
In abelian groups: $$\text{soc}\left(\prod_{p\text{ prime}} \mathbb{Z}/p\mathbb{Z}\right) = \bigoplus_{p\text{ prime}} \mathbb{Z}/p\mathbb{Z}\not\cong \prod_{p\text{ prime}}\mathbb{Z}/p\mathbb{Z} = \...
- 5,031
8
votes
Accepted
How to understand adjoint functors?
Nice question Bumblebee. So, let us start with some "metaphysics of adjointness":
THE LEFT AND RIGHT ADJOINTS TO A FUNCTOR
$ \mathcal{F}:\mathcal{C}\hookrightarrow\mathcal{D}$
ARE THE FREE (...
- 7,893
8
votes
A specific property of bi-adjunction
Qiaochu Yuan’s answer excellently explains the general phenomenon. Here is another slightly simpler concrete example:
Let $\newcommand{\Z}{\mathbb{Z}}\newcommand{\BZ}{{\mathbf{B}\Z}}\BZ$ be the group ...
- 21.5k
Only top scored, non community-wiki answers of a minimum length are eligible