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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 non-simplicial 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 non-simplicial model categories
deleted 111 characters in body
Aug
11
comment Quasicategories for non-simplicial 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 non-simplicial 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 topos-theoretic 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?