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Nov
25
comment Could the concept of “finite free groups” be possible?
The only free group that is finite is the trivial group. What on earth could a "finite free group" be?
Nov
25
comment colimit on presheafs
Or otherwise restrict to finite connected diagrams. Then it is true: the colimit of a finite connected diagram can indeed be constructed by iterated pushouts. The proof is ugly, however.
Nov
25
comment colimit on presheafs
What you claim cannot possibly be true: how do you use just pushouts to construct the colimit of a diagram with more than one connected component?
Nov
25
comment Definition of a one-connected manifold?
I have seen "1-connected" as an abbreviation for "simply connected".
Nov
25
comment Examples of canonical projections that are not epimorphisms and canonical injections that are not
Your claim is incorrect: the projection $X \times \emptyset \to X$ is surjective if and only if $X = \emptyset$.
Nov
24
comment What is homotopy in $(\infty,1)$-categories (as weak Kan complexes)
You basically choose two faces of the $(n + 1)$-simplex to be the source and target and require the remainder to be degenerate in the appropriate sense. The case $n = 1$ is discussed in any introduction to quasicategories.
Nov
24
comment What is homotopy in $(\infty,1)$-categories (as weak Kan complexes)
A homotopy between $n$-cells in a quasicategory is a special kind of $(n + 1)$-cell.
Nov
24
awarded  Nice Answer
Nov
24
answered What are presentable categories?
Nov
23
comment Are these categories toposes?
None of the above: all of these categories are not even cartesian closed.
Nov
22
comment Exact sequence of sheaves with non exact sequence of global sections
What's wrong with the exponential sequence as an example?
Nov
22
comment Is every complex (smooth) manifold a scheme?
Yes: schemes are by definition algebraic.
Nov
22
comment Is every complex (smooth) manifold a scheme?
The unit disc in $\mathbb{C}$ is surely a Riemann surface but not a complex variety. (Note, it is not isomorphic to $\mathbb{C}$ as a Riemann surface!)
Nov
22
comment Why are there no naturality condition in definition of exponential in a category?
Awodey's formulation includes the $\epsilon$, which implies naturality.
Nov
22
comment Endofunctor as a presheaf
We usually only consider presheaves over small categories. $\mathbf{Set}$ is not small.
Nov
21
comment On the notion of pro-category
The concept is defined in any category. The only question is whether it coincides what we normally think of as finitely presentable.
Nov
21
comment On the notion of pro-category
Surely you know what a finitely presentable group/module/etc. is? The abstract definition generalises those examples.
Nov
21
comment short exact sequences and functors
No. Why would we bother with the name "half-exact functor" if it was automatic?
Nov
21
comment On the notion of pro-category
Not quite. They are the ind-completion of the finitely presentable objects. Such categories have some of the properties familiar from algebra, e.g. filtered colimits preserve finite limits.
Nov
21
answered On the notion of pro-category