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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

10 votes
3 answers
1k views

Are inclusions "canonical" injections?

[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question] Summary of question: the inclusions are a particularly …
Kevin Buzzard's user avatar
22 votes
5 answers
2k views

are quotients by equivalence relations "better" than surjections?

This might be a load of old nonsense. I have always had it in my head that if $f:X\to Y$ is an injection, then $f$ has some sort of "canonical factorization" as a bijection $X\to f(X)$ followed by an …
Kevin Buzzard's user avatar
37 votes
Accepted

Is "all categorical reasoning formally contradictory"?

Note: I am not a historian. I'm just guessing as to what prompted the comments. Here's my guess: if you do set theory naively, in the old-fashioned "anything is a set" way, then you run into Russell' …
Kevin Buzzard's user avatar
73 votes

How do I check if a functor has a (left/right) adjoint?

Lots of people-who-are-fond-of-adjoint-functor-theorems have responded to this post saying "adjoint functor theorems". Let me give a more mundane and rather different answer which fits much better int …
Kevin Buzzard's user avatar
3 votes

Splitting lemma under assumption of the axiom of choice

The mistake you're making is that your map $u$ is not a homomorphism of (whatever $A$, $B$, $C$ are---possibly groups or modules), it's just going to be a map of sets, if you define it the way you def …
Kevin Buzzard's user avatar