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Dec
17
comment Pullbacks and the power set functor
With $\mathcal{P}_* D$ you mean the functor composition of $\mathcal{P}$ and $D$?
Nov
18
comment Taking the automorphism group of a group is not functorial.
Sorry Servaes, maybe I am missing something. Are you saying that $Aut$ works functorially almost everywhere in Grp except for some exotic group/group morphism that you cannot even remember right now?
Nov
18
comment If $T$ has an adjoint, why is $T(X\times Y)\simeq T(X)\times T(Y)$?
You start well Camilla, but you do not use the bonus universal arrows (unit and counit) that come with the adjunction. Stefan does that in his answer. Please note that your $\varphi$ is normally described as $\varphi^{-1}$ in the literature (you wrote the isomorphism in the opposite direction to the standard one). Please see CWM or Awodey. So your $\operatorname{rad}f$ is actually called a left adjunct $\operatorname{lad}f$ in the literature.
Nov
18
comment Definitions of adjoints (functional analysis vs category thy)
Andreas' answer is excellent and you can read some more in the "Notes" to the "Adjoints" chapter in CWM.The connection initially was just notational. Even the link to universal properties was not too clear, it seems. Any deeper connection comes much, much later (see @MartinBrandenburg 's link). If you are a beginner in CT, Andreas' answer should suffice. The widespread existence of the adjunction phenomenon is a marvelous discovery as it is, the naming is secondary. It could have been called transposition or conjugation or anything else giving the idea of an (almost) symmetric relation.
Nov
17
awarded  Yearling
Nov
16
comment ETCS set theory: Are empty sets isomorphic?
@ZhenLin so the category (or categories) describef by ETCS is equivalent to SET, but not necessarily isomorphic? Having more than one empty set has no weird consequences?
Nov
15
comment ETCS set theory: Are empty sets isomorphic?
@Svinepels that was the reason of my question. I am not very familiar with ETCS, but i would be very surprised to find out that it allows more than one empty set, if it has any pretense of mimicking the category of sets. So try to prove that there is exactly one empty set.
Nov
14
comment ETCS set theory: Are empty sets isomorphic?
is there more than one empty set?
Nov
8
comment Showing MON and CAT are equivalent categories
Mon and Cat are isomorphic or just equivalent?
Oct
28
comment Visualizing a homotopy pull back
Modern...yet Classical !?! sounds cool. Thank you
Oct
28
comment Visualizing a homotopy pull back
Welcome to Mathematics.SE! Are the notes for your course publicly availabel online?
Oct
27
comment Category of natural numbers with divisbility?
You don't want $0$ in this category. So the objects should be: the positive natural numbers $N^{+}$. Not the non-negative natural numbers as you state
Oct
27
comment Do adjoint functors really define monads?
@student I just edited a small typo. No math or meaning. Anything else what edited by someone else.
Oct
27
comment The empty function and constants
@Stefan your objection is correct. The problem stems from the fact that wikipedia is slighty wrong. Wikipedia should not write that the empty product is equal to....(using the equality sign), because the cartesian product is actually defined up to isomorphism and not by an equality. In other words: the equal signs that you see in the wikipedia section on nullary cartesian products should be replaced with isomrphism signs. To recap: $f \neq g$ so $A^{\emptyset} \neq B^{\emptyset}$ , but they are both isomorphic to singleton sets
Oct
27
revised The empty function and constants
edited small typos
Oct
27
suggested approved edit on The empty function and constants
Oct
27
answered Check a theorem about the category Set
Oct
27
comment Do adjoint functors really define monads?
@student Where exactly in Weibel are you looking at?
Oct
27
comment Do adjoint functors really define monads?
@GiorgioMossa the OP is interested in the comonad
Oct
27
comment Do adjoint functors really define monads?
Exactly! As a matter of fact, the same explanation appears in the first edition of CWM too.