Intereting Posts

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Axiom of choice, non-measurable sets, countable unions
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Use implicit function theorem to show $O(n)$ is a manifold
showing that the sequence $a_n=1+\frac{1}{2}+…+\frac{1}{n} – \log(n)$ converges
eigen values and eigen vectors in case of matrixes and differential equations
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Introductory book for homotopical algebra
REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?

Let $\mathcal E$ be a topos as in Mac Lane and Moerdijk.

A initial object in $\mathcal E$ can be obtained as the domain of the equalizer of the morphisms $P!,\epsilon P1:P1\to P^31$, where $1$ is a terminal object, $!:P^21\to 1$ is the unique morphism and $\epsilon$ is the counit of the adjunction $P^{op}\dashv P$.

Let $A$ be a object in $\mathcal E$.

How to construct explicity the unique morphism $0\to A$?

- Additive category that is not abelian
- Mathematical structures
- Concrete description of (co)limits in elementary toposes via internal language?
- What's the explicit categorical relation between a linear transformation and its matrix representation?
- Monos in $\mathsf{Mon}$ are injective homomorphisms.
- Category Theory: homset preserves limits

- What is category theory useful for?
- Learning Combinatorial Species.
- what are the product and coproduct in the category of topological groups
- Why are continuous functions the “right” morphisms between topological spaces?
- Long exact sequence into short exact sequences
- What is the categorical diagram for the tensor product?
- An arrow is monic in the category of G-Sets if and only if its monic the category of sets
- Inductive vs projective limit of sequence of split surjections
- Uniqueness of adjoint functors up to isomorphism
- The projective model structure on chain complexes

The method used by Mac Lane and Moerdijk (after Paré) is very elegant and efficient, but also rather abstract. Zhen Lin has given a nice response within this abstract framework.

If the method seems *too* abstract (i.e., hard to visualize or intuit), there are alternatives. One is to develop some internal logic from scratch, defining operators $\Rightarrow$ and $\forall$ on subobject lattices and then defining an internal intersection operator $P P A \to P A$ by writing down a formula for the intersection in terms of logical operators. Then define the initial object as the subobject of $1$ that is classified by the composite

$$1 \stackrel{t_{P 1}}{\to} P P 1 \stackrel{\bigcap}{\to} P 1$$

whose meaning is “take the intersection of the family of all subobjects of $1$”. This at least produces the minimal subobject $0$ of $1$. Now let $A$ be any object, and take the pullback of the singleton map $\sigma_A: A \to P A$ along the composite map $0 \to 1 \stackrel{t_A}{\to} P A$. Call the pullback $P$; then the map $P \to 0$ is a pullback of the monic $\sigma_A$, hence itself monic, hence an iso because $0$ is the minimal subobject of $1$ by construction. This means we have a map $0 \cong P \to A$ (where the map $P \to A$ is the other pullback projection).

This point of view can be found in some notes I wrote up, here.

The (co)monadicity result implies that, for all objects $X$ in $\mathcal{E}$, $\epsilon_X : X \to P^2 X$ is the equaliser of $\epsilon_{P^2 X}, P^2 \epsilon_X : P^2 X \to P^4 X$. Thus, to construct a morphism $0 \to X$, it is enough to find a morphism $f : 0 \to P^2 X$ such that $\epsilon_{P^2 X} \circ f = P^2 \epsilon_X \circ f$. Consider $P ! : P 1 \to P^2 X$ and $P^3 ! : P^3 1 \to P^4 X$. By naturality, $P^3_! \circ \epsilon_{P 1} = \epsilon_{P^2 X} \circ P !$, and by functoriality, $P^3 ! \circ P ! = P^2 \epsilon_X \circ P !$. Thus, we may take $f : 0 \to P^2 X$ to be the composite $0 \to P 1 \stackrel{P !}{\to} P^2 X$, and this induces the required morphism $0 \to X$.

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