26 votes

Are there non-trivial infinite chains of adjoint functors?

Let $C$ be a category enriched over finite-dimensional $k$-vector spaces. A Serre functor for $C$ is a $k$-linear automorphism $S : C \to C$ such that there is a natural equivalence $$\text{Hom}(x, y)...
21 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) \...
20 votes

Are there non-trivial infinite chains of adjoint functors?

Broadening the question a bit, you can ask the same question about adjoint 1-morphisms in a 2-category (you're asking about 1-morphisms in the 2-category of categories). Then the 2-dimensional framed ...
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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-...
19 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,...
  • 52.8k
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 ...
18 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.
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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 ...
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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 ...
  • 30.1k
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 $G$...
14 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, ...
  • 10.2k
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 ...
  • 26.1k
12 votes
Accepted

On the coherence theorem for bicategories

Probably you had some trouble finding this because the search term $2$-$\text{Cat}$ is not accurate enough; you want not the cartesian monoidal product on $2$-$\text{Cat}$, but rather what is called ...
  • 51k
12 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 ...
  • 52.8k
12 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 ...
11 votes
Accepted

Why quasi-inverse functors are adjoint pairs?

This is a well-known result in category theory: that any equivalence can be improved to an adjoint equivalence. Given an equivalence in the form of isomorphisms $\eta: 1_C \stackrel{\sim}\to GF$ and $\...
  • 51k
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_{\...
  • 15.8k
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 ...
  • 35.1k
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 ...
  • 52.8k
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]\...
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 ...
  • 52.8k
9 votes

Uniqueness of $\infty$-adjoints

Yes, and $\mathrm{Adj}_\infty = \mathrm{Adj}_1$. This is Theorem 4.4.18 of Riehl-Verity. Thanks David, I should have remembered that.
9 votes

Are there non-trivial infinite chains of adjoint functors?

Another widely used example of infinite adjunction chains arises in linguistics, specifically in connection with pregroup grammars. It has been first observed by Lambek in Some Galois Connections in ...
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

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 ...
  • 52.8k
9 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 ...
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^{\...
  • 10.2k
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 $\...
8 votes

Are there non-trivial infinite chains of adjoint functors?

Another nice example is the infinite sequence of adjunctions characterizing stable homotopy theories. One has, in any homotopy theory $K$ (whatever you think that is, as long as there is a notion of (...
  • 2,172
8 votes

Lurie's approach to the bar-cobar adjunction

I really like this question, I've been trying to sort out some of these ideas for a little while. I don't know the answer to your questions about conilpotence and twisting morphisms vs twisted arrows. ...

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