Re: Lawvere's characterization of adjunctions

Re: Lawvere's characterization of adjunctions

[Colin McLarty]

[-- Attachment #1: Type: text/plain, Size: 2102 bytes --]


It depends on what you want to tell people.  The idea per se is very simple.  Once you have it, you see it is just exactly what every definition of adjoint functors says, put in arrow theoretic terms. .

For references that typify Bill's thinking about it see his dissertation and his Category of Categories as Foundations.

For a concise, authoritative treatment see Mac Lane's exercise 2 of Section IV.1 Adjoints, in Categories for the Working Mathematician 2nd ed.  (It is also in the first ed but numbered differently and I do not have that on hand.)

For a leisurely account using this idea from the ground up, see my book Elementary Categories, Elementary Toposes.

Colin



On Sat, Jun 21, 2025 at 7:25 AM Uwe Egbert Wolter <Uwe.Wolter@uib.no<mailto:Uwe.Wolter@uib.no>> wrote:
Dear all,

What reference you would recommend for Lawvere's characterization of adjunctions by means comma categories?

Best regards, Uwe


You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message.

View group files<https://url.au.m.mimecastprotect.com/s/-F1cCyoj8PurjkpzGTZfLhxBnWs?domain=outlook.office365.com>   |   Leave group<https://url.au.m.mimecastprotect.com/s/Vu-vCzvkmpfMP97pXiXhQh9NGWX?domain=outlook.office365.com>   |   Learn more about Microsoft 365 Groups<https://url.au.m.mimecastprotect.com/s/zvp0CANpnDCNm283xT9i2hGhlO7?domain=aka.ms>



You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message.

View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Learn more about Microsoft 365 Groups<https://aka.ms/o365g>


[-- Attachment #2: Type: text/html, Size: 6759 bytes --]

Re: Lawvere's characterization of adjunctions

[George Janelidze]

[-- Attachment #1: Type: text/plain, Size: 1691 bytes --]

Dear Uwe,

I would suggest to decide how to talk about this after looking at:


  1.  Two places in the TAC reprint of Lawvere’s “FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES AND SOME ALGEBRAIC PROBLEMS IN THE CONTEXT OF FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES”, namely: (i) Item (2) (Page 8) of “1. Seven ideas introduced in the 1963 thesis”; (ii) A part of “2. Adjoint functors” on Pages 38-41.
  2.  Exercise 2 of Section 1 of “IV. Adjoints” from Mac Lane’s “CATEGORIES FOR THE WORKING MATHEMATICIAN” (It is on Page 84 in the 1st Edition, and on Page 86 in the 2nd Edition).

Best regards, George

From: Uwe Egbert Wolter <Uwe.Wolter@uib.no>
Date: Monday, 23 June 2025 at 10:11
To: categories@mq.edu.au <categories@mq.edu.au>
Subject: Lawvere's characterization of adjunctions
CAUTION: This email originated outside the UCT network. Do not click any links or open attachments unless you know and trust the source.

Dear all,

What reference you would recommend for Lawvere's characterization of adjunctions by means comma categories?

Best regards, Uwe


You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message.

View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Learn more about Microsoft 365 Groups<https://aka.ms/o365g>


[-- Attachment #2: Type: text/html, Size: 8926 bytes --]