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Mar
28
comment Characterization of fully faithful functors as objects in a functor category
Obviously not. For instance, suppose all of the categories involved are actually sets. Then fully faithful functors are the same as injective maps, so you are asking if bijections of function-sets preserve injectivity.
Mar
24
comment How, intuitively, does commuting with filtered colimits capture “smallness”?
Of course, it is worth pointing out that compact topological spaces are not literally compact objects in $\mathbf{Top}$...
Mar
24
comment Intuitive meaning for Kan fibration
It is a morphism that has a combinatorial version of the homotopy lifting property.
Mar
23
comment on the necessity of gluing conditions
Well, it depends on what you expect to get after you glue. Suppose the result is $X$; by abuse of notation, consider each $X_i$ as a subobject of $X$; then one would expect $X_i \cap X_j = U_{i,j} = U_{j,i}$.
Mar
22
comment Effective equivalence relations in a topos
I don't think so. The isomorphism should involve both symmetry and transitivity.
Mar
22
comment Effective equivalence relations in a topos
Have you tried to see what happens in $\mathbf{Set}$?
Mar
18
comment Smooth completion of algebraic curves
Yes, that's one way of doing it. Blowups are birational and every birational equivalence class of projective algebraic curves contains a unique smooth algebraic curve.
Mar
18
comment Smooth completion of algebraic curves
Even in that case it can happen that the projective closure is not smooth.
Mar
18
comment Smooth completion of algebraic curves
Given a smooth (affine) algebraic curve embedded in some affine space, its projective closure can fail to be smooth. The correct procedure is described in §6 of [Hartshorne, Ch. I].
Mar
17
comment Can some Lie groups ($S^3$ in particular) be converted to simplicial groups?
The singular simplicial set functor takes topological groups to simplicial groups.
Mar
14
comment Defining Presheaves on Categories
Actually, you don't need to assume the existence of limits. The sheaf condition can be restated in the form "<a certain cone> is a limiting cone".
Mar
14
comment Defining adjoint functors: What does “natural bijection” mean?
The second sentence is just repeating what the first sentence is saying in more basic terms. If you unfold the definition of natural isomorphism you will see this.
Mar
14
comment $(R^{\oplus A})^{\oplus B} \approx R^{\oplus (A\times B)}$?
You need to use the fact that, for sets, multiplication really is just repeated addition.
Mar
14
comment Square is homotopy Cartesian if horizontal maps are weak equivalences
It's actually true in any model category (or even any category with weak equivalences) but the proof would be different.
Mar
9
comment References about positivism
Isn't there something called positive set theory?
Mar
9
comment Constructing natural numbers as lists of units (possible infinite objects)
The standard definition of list types does not admit infinite lists. It seems to me that you are confusing inductive data structures with coinductive data structures.
Mar
7
comment Which ZF axioms are satisfied in $V_{\omega+\omega}$?
Is $\omega + \omega$ a set in $V_{\omega + \omega}$? Can you make it using replacement?
Mar
7
comment Can we simultaneously freely adjoin both limits and colimits to a category?
This is described in Joyal's paper to some degree, but it's not very explicit.
Mar
7
comment Can we simultaneously freely adjoin both limits and colimits to a category?
It does not seem to be available online.
Mar
7
comment Why is $Set$ not equivalent to $Set^{op}$?
The second paragraph is the explanation. Try dualising the two properties in question.