Intereting Posts

Where, specifically, did Principia Mathematica fail?
how many permutations of {1,2,…,9}
Why is the empty set a subset of every set?
How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?
How to calculate the pullback of a $k$-form explicitly
Does a injective function $f: A \to B$ and surjective function $g : A\to B$ imply a bijective function exists?
Proving this formula $1+\sum_{n=0}^{\infty }\frac{1}{\pi \left(2n+\frac{3}{4}\right)\left(2n+\frac{5}{4}\right)}=\sqrt2$
Missing term in series expansion
Tensor product of monoids and arbitrary algebraic structures
Confusing Trigonometry Problem
Prove $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$
Are the events independent?
Can every curve on a Riemannian manifold be interpreted as a geodesic of a given metric?
How to integrate $\int \frac{1}{\sin^4x + \cos^4 x} \,dx$?
If a rational number has a finite decimal representation, then it has a finite representation in base $b$ for any $b>1?$

According to the nLab entry for abstract cobordism categories, the natural way of axiomatizing the relation of two oriented manifolds being cobordant is the following:

Definition 1Two objects $M,N$ in a cobordism category $(D,\partial,i)$ are said to be cobordant if there are objects $U,V\in D$ such that $M+\partial U \simeq N+\partial V$.

My question is how to reconcile this definition with the classical definition of oriented cobordism:

- In (relatively) simple words: What is an inverse limit?
- Tensor products from internal hom?
- What is a natural isomorphism?
- Category Theory vs. Universal Algebra - Any References?
- A Question on a claim regarding the notion of “space” in “Indiscrete Thoughts”
- Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?

Definition 2Two $n$-dimensional manifolds $M,N$ are said to be cobordant if there is an $n+1$-dimensional manifold $W$ such that $\partial W\simeq \bar{M}+N$. Here $\bar{M}$ is defined as $M$ after an orientation reversal.

The problem only seems to arise when $M$ and $N$ are not cobordant to the empty manifold.

- Coproduct in the category of (noncommutative) associative algebras
- Realizing the monoid $\mathbb{N}/(3=1)$ from a category with finite coproducts
- $F$ is an equivalence of categories implies that $F$ is fully faithful and essentially surjective
- coproducts of structures
- What are the epimorphisms in the category of Hausdorff spaces?
- Why the morphisms of vector spaces, over different fields is not interesting?
- Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes
- Fibered coproducts in $\mathsf{Set}$
- Is there a monoid structure on the set of paths of a graph?
- Left Adjoint of a Representable Functor

These two definitions of oriented cobordism are equivalent (it’s a kind of a variation on the theme $a-b=0\iff a+x=b+x$).

One direction is rather obvious: $\partial W\cong\bar M+N\implies M+\partial W=N+\partial(M\times[0;1])$.

The opposite direction is slightly more interesting. Observe that $A+\partial B\cong\partial C\implies \exists W:A\cong\partial W$ (just take $W=C\sqcup_{\partial\bar B}\bar B$). So $M+\partial U\cong N+\partial V\implies \partial(M\times[0;1]+ U)\cong(\bar M+N)+\partial V\implies\bar M+N\cong\partial W$.

- How to test whether a set of four points can form a parallelogram
- limit of reciprocal of function
- Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is…
- Calculate probability density function from moment generating function
- Calculating Christoffel symbols using variational geodesic equation
- Finding XOR of all even numbers from n to m
- The Greatest Common Divisor of All Numbers of the Form $n^a-n^b$
- Area and Polar Coordinates
- Calculate $\mathrm{Gal}(\mathbb{Q}(\sqrt{3})/\mathbb{Q})$
- Are Jordan chains always linearly independent?
- Closed form for $\int_0^{\infty}\sin(x^n)\mathbb{d}x$
- Elementary Set Theory Question
- Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$
- Sum of every $k$th binomial coefficient.
- Trapping region for Nonlinear ODE system?