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1d

revised 
I discovered something, what do I do?
edited tags 
1d

comment 
Can such categorical notion of action be formalized?
It is not as general as you ask for. I am just pointing out that there is something that captures the examples you mention. 
1d

comment 
Can such categorical notion of action be formalized?
These are all essentially examples of monoid actions for a monoid in a monoidal category. 
2d

comment 
Kan extensions of pseudofunctors
It depends on what you mean. As Finn Lawler explained, you only need (pseudo)colimits of diagrams of shape $\mathcal{A}$ weighted by certain (pseudo)functors. Whether you can reduce to conical (pseudo)colimits or not depends on the weights – specifically, whether the weights themselves can be reduced to conical (pseudo)colimits of representables. 
2d

comment 
Kan extensions of pseudofunctors
The "standard" formula doesn't generalise well. You would be better off starting with the formula from enriched category theory in terms of weighted colimits. 
Aug
28 
comment 
Which sequential colimits commute with pullbacks in the category of topological spaces?
I guess you are referring to the proof here? It doesn't really indicate what the axioms are supposed to be, but I gather that they are based on ultrafilter convergence in $\Delta^n$ or something like that. That is why I believe the category is locally $(2^{\aleph_0})^+$presentable, but as I said, I haven't seen a complete proof. 
Aug
28 
comment 
Which sequential colimits commute with pullbacks in the category of topological spaces?
I'm not sure I believe that the category of $\Delta$generated spaces is locally $\lambda$presentable for $\lambda \ge 2^{\aleph_0}$. I might believe $\lambda > 2^{\aleph_0}$, but it's still something I haven't seen proved. 
Aug
28 
comment 
Which sequential colimits commute with pullbacks in the category of topological spaces?
You need a locally finitely presentable category for this. An $\omega$sequence is not $\kappa$filtered for any uncountable regular cardinal $\kappa$. I know the category of $\Delta$generated spaces is locally presentable, but I doubt it is locally finitely presentable. 
Aug
27 
comment 
A “universally non Hypercomplete” $\infty$topos?
Tangentially, what does the $\infty$topos of local systems of spectra classify? 
Aug
21 
answered  Algebras for probability monad 
Aug
18 
revised 
(infinity,1)categories directly from model categories
updated arxiv link 
Aug
14 
comment 
Grothendieck toposes in (very) weak foundation
We can think about universal closure operators instead of local operators. 
Aug
11 
comment 
Quasicategories for nonsimplicial model categories
Something like that, yes. As you remarked, left proper + weak equivalences closed under coproducts implies there is a maximal/canonical cofibration structure. 
Aug
11 
revised 
Quasicategories for nonsimplicial model categories
deleted 111 characters in body 
Aug
11 
comment 
Quasicategories for nonsimplicial model categories
Sure, of course. I was thinking of using the model category itself as a category of cofibrant objects, but I guess either way the vertices are still going to be more complicated than just objects of the original category. 
Aug
11 
answered  Quasicategories for nonsimplicial model categories 
Aug
10 
comment 
Categories of finite objects
@TomBachmann Compact objects in the (∞, 1)category of spaces are more general than finite CW complexes, however. 
Aug
7 
comment 
Does this axiom (a weak form of class valued choice) has a name?
I think this is a variation of (algebraic set theory version of) the axiom of collection. See [Joyal and Moerdijk, Algebraic set theory, Ch. I, §1]. 
Aug
4 
comment 
Does a topological hypercover always have free degeneracies?
@JesseC.McKeown That is correct. In topostheoretic language, my argument amounts to the observation that a sheaf whose espace étalé is Hausdorff is a decidable object, i.e. $A \times A \cong \Delta_A \amalg B$, where $\Delta_A$ is the diagonal; a simplicial sheaf that is degreewise decidable will then have free degeneracies. 
Aug
4 
answered  Does a topological hypercover always have free degeneracies? 