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Jan
28
comment Adjoint to $\mathsf{Proj}$? - A quest to understand categories of graded objects.
Incidentally, the business with removing the origin is precisely why $\mathrm{Proj}$ is not functorial (with respect to graded ring homomorphisms).
Jan
28
comment Is an equivalence an adjunction?
They would have to satisfy the triangle identities if they were unit and counit for the same adjunction.
Jan
28
comment When is a functor of bicategories part of an equivalence?
Yes, there will be a pseudofunctor quasi-inverse. It's briefly mentioned in [Lack, A 2-categories companion].
Jan
28
comment Is an equivalence an adjunction?
Yes, $\epsilon$ is a counit, but not necessarily for the same adjunction.
Jan
27
comment When is a functor of bicategories part of an equivalence?
It is in fact equivalent to the axiom of choice.
Jan
27
comment What does “variance unity” mean?: “A normal distribution with mean $\mu$ and variance unity”.
"Unity" is sometimes a synonym for the number 1.
Jan
26
comment Matching faces in Simplicial Set theory
Yes, adjacent is a good word.
Jan
26
comment Matching faces in Simplicial Set theory
They match along lower-dimensional faces. Try drawing a picture for $n = 2$ or $n = 3$.
Jan
26
comment Product of compacts is compacts using closedness of projection to second component?
Everything in my post goes through for not-necessarily-continuous closed maps.
Jan
26
comment Product of compacts is compacts using closedness of projection to second component?
There are variations in the terminology. Here, I mean a closed (continuous) map such that the inverse image of a compact subspace is compact. It is the same as what Stefan Hamcke calls "perfect".
Jan
25
comment The étale fundamental group as a functor
You should be able to find in standard references that the class of finite (resp. étale, surjective) morphisms is closed under pullback. For instance, try the Stacks project.
Jan
25
comment Awodey's Category Theory Exercise 9.9.2
The monoid homomorphism $F (X) \to M$ you should be considering is not arbitrary. The hint says to think about the adjunction counit. Well, it is a monoid homomorphism – did you try showing it is surjective?
Jan
25
comment Awodey's Category Theory Exercise 9.9.2
For this exercise, I do not think it is helpful to think of monoids as being special categories. If you insist, "surjective homomorphism" here means "full functor".
Jan
25
comment Awodey's Category Theory Exercise 9.9.2
Well, actually, the question says "surjection", so why don't you prove that it's surjective? It will then follow that you have an epimorphism.
Jan
23
comment Proper maps in terms of projection from pullback
(1) The proof is not obvious even in the special case where $S$ is a point. It involves some carefully chosen topological spaces. See tag 005M in [Stacks].
Jan
23
comment The category of (completable) rings has enough projectives in it
Cauchy completion (of metric spaces) is not functorial with respect to continuous maps though, so you'll also have to say what the morphisms in your category are...
Jan
23
comment Proper maps in terms of projection from pullback
What's your definition of proper map if not this?
Jan
23
comment Homotopy colimit,weighted colimit, homotopy theory
Take $m = \operatorname{hocolim} F$ on the LHS. Then you have $\mathrm{id}$. Follow it across the bijection to get the desired universal simplicial natural transformation.
Jan
19
comment Scheme morphism properties that aren't stable under taking triangles?
It has very little to do with the empty set. It's just an extreme case – you can easily conjure up examples with non-empty domain. What's more interesting is to find ones where $g$ is surjective...
Jan
19
comment Scheme morphism properties that aren't stable under taking triangles?
There are lots of examples. For instance, suppose every morphism $\emptyset \to X$ is in $\mathcal{P}$. Then the two-out-of-three condition would imply every morphism is in $\mathcal{P}$ – but this is plainly not true when e.g. $\mathcal{P}$ is the class of open immersions, étale morphisms, affine morphisms, morphisms of finite presentation...