Intereting Posts

the number of the positive integer numbers $k$ that makes for the two quadratic equations $ \pm x^2 \pm px \pm k$ rational roots.
Soberification of a topological space
Penrose tilings as a cross section of a $5$-dimensional regular tiling
In a P.I.D., if $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$ with $\gcd(m,n) = 1$, then $a=b$
The center of the dihedral group
Integrating $e^{a\cos(\phi_1-\phi_2)+b\cos(\phi_1-\phi_3)+c\cos(\phi_2-\phi_3)}$ over $^3$
In Bayesian Statistic how do you usually find out what is the distribution of the unknown?
Finite Extensions and Roots of Unity
Distributing $n$ different things among $r$ distinct groups such that all of them must get atleast $1$
Prove that if $ab \equiv 1 \pmod{p}$ and $a$ is quadratic residue mod $p$, then so is $b$
What kind of book would show where the inspiration for the Laplace transform came from?
How can I calculate closed form of a sum?
Parameters on $SU(4)$ and $SU(2)$
Throwing $k$ balls into $n$ bins.
How many entries in $3\times 3$ matrix with integer entries and determinant equal to $1$ can be even?

Every pair $F \dashv G$ of adjoint functors $F: \mathcal C \to \mathcal D$, $G: \mathcal D \to \mathcal C$ induces a monad $\mathbb T = (T,\eta,\mu)$ on $\mathcal C$. Given a monad $\mathbb T = (T,\eta,\mu)$ on $\mathcal C$, we define $\operatorname{Adj}(\mathbb T)$ to be the category of adjunctions inducing $\mathbb T$. Its objects are adjoint pairs of functors $F \dashv G$ between $\mathcal C$ and some category $\mathcal D$ such that the monad induced by the adjunction is $\mathbb T$, i.e. $GF = T$, the unit is $\eta$, and $G\varepsilon F = \mu$ where $\varepsilon$ is the counit. The morphisms between two such adjunctions $F \dashv G$, $F’ \dashv G’$ are functors $H: \mathcal D \to \mathcal D’$ such that $HF=F’$ and $G’ H = G$.

I am now worried whether this is really a category. Consider the example where $\mathcal C$ is a one-object category with only the identity morphism and $\mathbb T = (T,\eta,\mu)$ is the trivial monad. For every category $\mathcal D$ which has an initial object $I$, we get an adjunction $F \dashv G$, where $F: \mathcal C \to \mathcal D$ sends the one object of $\mathcal C$ to $I$, and $G: \mathcal D \to \mathcal C$ sends everything to the one object (it’s easy to check that this is really an adjunction). Its induced monad on $\mathcal C$ is necessarily $\mathbb T$. So $\operatorname{Adj}(\mathbb T)$ contains at least one object for each category having an initial object. However, $\operatorname{Adj}(\mathbb T)$ itself has an initial object, namely the Kleisli adjunction, which in this case is just $1_{\mathcal C} \dashv 1_{\mathcal C}$. If $\operatorname{Adj}(\mathbb T)$ is a category, it seems that in a way it contains itself, in the form of the adjunction $\mathcal C \rightleftarrows \operatorname{Adj}(\mathbb T)$ defined by the Kleisli adjunction. (How) Is that possible?

- Why are topological spaces interesting to study?
- Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers
- What Is a Morphism?
- Equivalence of the definition of Adjoint Functors via Universal Morphisms and Unit-Counit
- Universal property of tensor products / Categorial expression of bilinearity
- Quantificators vs pullbacks

- Proof of Yoneda Lemma
- Yoneda implies $\text{Hom}(X,Z)\cong \text{Hom(}Y,Z)\Rightarrow X\cong Y$?
- What's the best way to teach oneself both Category Theory & Model Theory?
- Are all projection maps in a categorical product epic?
- Equivalent definitions of regular categories?
- Natural isomorphisms and the axiom of choice
- Can finding a left-inverse be formulated as a problem of choosing from a set?
- Can a free group over a set be constructed this way (without equivalence classes of words)?
- How to define Homology Functor in an arbitrary Abelian Category?
- Completion as a functor between topological rings

Well, either one works in a framework where self-containing categories are permitted (but I do not know of any), or else one works more carefully. So I would say something like, fix a universe $\mathbf{U}$ such that $\mathcal{C}$ is a $\mathbf{U}$-small category, and let $\mathbf{Adj}(\mathcal{C})$ be the category of $\mathbf{U}$-small categories equipped with an adjunction etc.; then $\mathbf{Adj}(\mathcal{C})$ will not be $\mathbf{U}$-small, and so will not member of itself.

- Equation of a rectangle
- What's wrong with this proof that $e^{i\theta} = e^{-i\theta}$?
- Use Little Fermat Theorem to prove $341$ is not a prime base $7$
- Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$
- Evaluating $\int_{0}^{1} dx\frac{\log(1+x)}{1 + x^2}$
- Show: $f'=0\Rightarrow f=\mbox{const}$
- How can we prove that among positive integers any number can have only one prime factorization?
- Formal proof that Schwartz space is space of rapidly decreasing functions
- Is there a $C^1$ curve dense in the plane?
- Defining the integers and rationals
- Today a student asked me $\int \ln (\sin x) \, dx.$
- Factor $x^4+1$ over $\mathbb{R}$
- Group of order $p^2+p $ is not simple
- A one-to-one function from a finite set to itself is onto – how to prove by induction?
- Bayesian posterior with truncated normal prior