Intereting Posts

What is the geodesic beween two opposite corners of cube on its surface?
If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality
Primality of $n! +1$
Is there no solution to the blue-eyed islander puzzle?
Showing that $n$ is pseudoprime to the base $a$
How to conceptualize unintuitive topology?
When does a null integral implies that a form is exact?
De Rham Cohomology of $M \times \mathbb{S}^1$
Prove $P_m X P_n$ is Hamiltonian if and only if at least one of $m,n$ is even
If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L<\infty$, does $\lim\limits_{x\rightarrow\infty} f(x) $ exist?
Countably monotone does not imply monotone and countably additive?
Evaluate $\int x^2e^{x^2} dx$
Why is $L_A$ not $\mathbb K$ linear (I can prove that it is)
Poisson-processes and it's arrival times
How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn(n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

Been reading up on the idea of distributive categories. Suppose $\mathcal{C}$ is some category such that for all $A,B\in\mathcal{C}$ the product $A\times B$ and coproduct $A\oplus B$ exist.

So $\mathcal{C}$ is a distributive category if the canonical morphism

$$

\phi\colon (A\times B)\oplus(A\times C)\to A\times (B\oplus C)

$$

is an isomorphism.

This is a basic question, but what precisely is this so called canonical morphism? Really, what does an arbitrary “thing” (not sure if element is the right word here) in $(A\times B)\oplus(A\times C)$ look like, and where does it go under $\phi$?

- definition of a free object in a category
- Describing the Wreath product categorically.
- Special arrows for notation of morphisms
- What is a natural isomorphism?
- What is the most general category in which exist short exact sequences?
- A push-out of a pull-back

- Axiomatizing oriented cobordism
- Regular monomorphisms of commutative rings
- ''Labelling discrimination'' for objects in a category
- Is this equality in a double category true?
- Coproducts in $\text{Ab}$
- Why is in the category of pointed sets not every epimorphism a cokernel?
- Do finite products commute with colimits in the category of spaces?
- How do you break up an exact sequence of any length to a “succession of short exact sequences”?
- What good are free groups?
- Equivalence of categories ($c^*$ algebras <-> topological spaces)

There is a canonical morphism $B\to B\oplus C$, which induces a morphism $\phi_1:A\times B\to A\times(B\oplus C)$. Similarly, there is a canonical morphism $C\to B\oplus C$ which induces a morphism $\phi_2:A\times C\to A\times(B\oplus C)$.

Now $\phi_1$ and $\phi_2$ determine a unique morphism $(A\times B)\oplus(A\times C)\to A\times(B\oplus C)$. That’s your morphism.

- How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?
- What is a complex inner product space “really”?
- Moriarty's calculator: some bizarre and deceptive graphical anomalies
- When is a metric space Euclidean, without referring to $\mathbb R^n$?
- How do I evaluate the limit $\lim_{n\to\infty}n((1+1/n)^n-e)$?
- What is the free category on the underlying graph of a category?
- Number of solutions to equation.
- Proving an invertible matrix which is its own inverse has determinant $1$ or $-1$
- Burnside's Lemma
- Number of ways to express a number as the sum of different integers
- Let $A \subseteq X$ and $f: X \mapsto X$. Prove $f^{-1}(A) = A \iff f(A) \subseteq A \land f^{-1}(A) \subseteq A$
- Levin's u-transformation
- Finding integers satisfying $m^2 – n^2 = 1111$
- 101 positive integers placed on a circle
- What does a “half derivative” mean?