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Feb
3
comment Name for categories with a certain property on coproducts
In some ways this is the complement of Bourn's notion of unital category, which asks for the canonical $X_0 + X_1 \to X_0 \times X_1$ to be a strong epimorphism.
Feb
1
comment Complete atomic boolean algebras as coalgebras of some endofunctor on Set
There are ways to prove such results, yes. For instance, the category of coalgebras for an endofunctor will be closed under retracts in a strong sense.
Feb
1
comment Complete atomic boolean algebras as coalgebras of some endofunctor on Set
Complete atomic boolean algebras are algebras for a monad. What makes you think they should be coalgebras for an endofunctor?
Feb
1
comment How do you know two morphisms are equal (without using elements)
Given two elements in a set, how can you know if they are equal?
Jan
31
comment Presheaves,simplicial sets, evaluation functor, Yoneda lemma,hom-functor
In fact this statement is just another way of stating the Yoneda lemma. You just have to unfold the definitions.
Jan
31
comment How do you define such map $(C^B \times B^A) \to C^A$?
Of course they are not literally equal. They don't even have the same domain and codomain.
Jan
30
comment A Grothendieck topology on $\Delta$
The Segal conditions are not really a sheaf condition – remember, covering sieves in a Grothendieck topology have to be closed under pullbacks. (In particular, neither quasicategories nor categories form a topos.)
Jan
29
comment The cofree coalgebra using adjoint functor theorems
In Q2, shouldn't you be looking for a weakly terminal set of objects?
Jan
29
comment The cofree coalgebra using adjoint functor theorems
I suppose it's possible in principle. If you know that $\mathbf{Coalg} (\mathcal{C})$ is $\kappa$-accessible then the $\kappa$-presentable objects form a dense subcategory. The problem is that, in the standard textbooks, no estimate of $\kappa$ is given – you have to chase through the proofs carefully. However, you might have better luck looking at the 1977 preprint of Ulmer (check your email).
Jan
29
comment How do you define such map $(C^B \times B^A) \to C^A$?
There's only one reasonable candidate: composition. And yes, you have to use evaluation and transposition.
Jan
28
comment Cohomology induces a functor
Yes. First one has to define cohomology in terms of just kernels and cokernels.
Jan
28
comment Can a Compact Lie Group have a Non-Compact Lie Subgroup?
Is it a topological embedding, however?
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 $n = 3$.