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Jan
14
answered Monoid and category
Jan
14
comment What is a codomain of diagonal functor?
@Denis It's not strange: it's an unavoidable fact that there are only finitely many short words in any language...
Jan
14
comment Set theory aspects of category theory
The existence of a universe containing $\omega$ in a universe satisfying ZFC implies ZFC is consistent, yes.
Jan
14
comment Set theory aspects of category theory
You could start by looking at the appendix of [Kelley, General topology].
Jan
14
comment Are discrete fibrations of monoids equivalent to monoid actions?
@Berci There is no notion of "equivalence" in $\mathbf{Set}$. So obviously the two categories are not equivalent.
Jan
13
revised Properties of $\pi_n$ from a category theoretical point of view
deleted 10 characters in body
Jan
13
comment Properties of $\pi_n$ from a category theoretical point of view
@Marek That's right: we do not have a nice algebraic model for $n$-groupoids as we do for 1-groupoids. We do have combinatorial models based on (truncated) simplicial sets, though.
Jan
13
awarded  Nice Question
Jan
13
answered Are discrete fibrations of monoids equivalent to monoid actions?
Jan
13
comment Is a nonzero divisor locally nonzero divisor?
It's also true if each $\mathscr{O}_{X, x}$ is flat over $\Gamma (X, \mathscr{O}_X)$.
Jan
13
comment Are discrete fibrations of monoids equivalent to monoid actions?
@Berci No. Let $\mathbb{A}$ be a "codiscrete" groupoid (= has exactly one (iso)morphism between any two objects). Then $\mathbb{A} \to \mathbb{1}$ is faithful.
Jan
13
answered Double negation distributes over conjunction
Jan
12
answered Properties of $\pi_n$ from a category theoretical point of view
Jan
11
comment Signatures having precisely one constant symbol, and pointed categories.
That would work.
Jan
11
answered Signatures having precisely one constant symbol, and pointed categories.
Jan
11
comment What are simplicial topological spaces intuitively?
That's not true. Simplicial sets admit (by design) a geometric realisation functor that give them their geometric meaning. I know of no such thing for simplicial spaces.
Jan
11
comment Are discrete fibrations of monoids equivalent to monoid actions?
Your claim is still incorrect. $\mathbf{Grpd}_{/ G}$, as you describe it, is not equivalent to $\mathbf{Set}^G$, not even for the trivial group $G$.
Jan
11
comment What are simplicial topological spaces intuitively?
This is quite down-to-earth, really: I'm saying a simplicial space is not much more than a diagram of spaces. There's nothing special about $\mathbf{\Delta}$ either – the same goes for the category of diagrams $\mathcal{C} \to \mathbf{Top}$ for any small category $\mathcal{C}$.
Jan
10
comment Commutativity of the square diagram coming from an adjoint triple
Yes, or more simply, by naturality.
Jan
10
answered What are simplicial topological spaces intuitively?