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Aug
30
comment A “universally non Hypercomplete” $\infty$-topos?
The localization can be described more concretely: it's sheaves with respect to the Grothendieck topology where every morphism generates a covering. Equivalently, it's functors F: {finite spaces} -> {spaces} (or you can do a pointed version if you like) with the property that for any X, F(X) is the totalization of the cosimplicial space given by applying F to the "Cech nerve" of the map X->* (in the opposite of finite spaces). This is much larger than the class of 1-excisive functors: it contains all n-excisive functors for any n, and more (such as products of n-excisive functors as n varies).
Aug
15
comment Schwede-Shipley theorem for monoidal categories?
True iff the unit for the monoidal structure is a compact generator.
Aug
11
comment Can hypercomplete objects be coreflective?
Not sure if it is explicitly stated there, but it is at least easily derived from what's there. First, reduced 1-excisive functors are spectra (Calculus III). If F is an arbitrary excisive functor then Y=F(*) is a space and $(y \in Y) \mapsto \lambda X. F(X) \times_{F(*)} \{y\}$ is a local system of reduced excisive functors on $Y$. Conversely if $\{ F_y \}_{y \in Y}$ is a local system of reduced excisive functors on a space Y, then $F(X)=\varinjlim_{y \in Y} F_y(X)$ is an excisive functor (not necessarily reduced). These constructions are homotopy inverse to one another.
Aug
10
comment Can hypercomplete objects be coreflective?
Tom Goodwillie, "Calculus III: Taylor Series"
Aug
6
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Aug
6
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Aug
6
revised Can hypercomplete objects be coreflective?
Added clarification about base points
Aug
6
answered Can hypercomplete objects be coreflective?
Jul
18
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Jul
17
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Jul
17
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Feb
20
comment Analogues of Primitive Recursive Functions
I'm afraid that I don't follow either. What exactly are you saying you can rule out?
Feb
19
awarded  Nice Question
Jan
28
comment Analogues of Primitive Recursive Functions
@Ulrik That paper looks quite relevant, but not exactly what I'm looking for. If I'm reading it right, it seems to be about functions whose totality is provable using very weak set-theoretic assumptions, analogous to the characterization of primitive recursive functions as those functions which are provably total using very weak arithmetic assumptions. But I'm hoping for something which is specific to a fixed admissible set $A$, and specializes to primitive recursive arithmetic when I take $A = HF$. (Something that would generalize PRA, rather than being analogous to PRA.)
Jan
28
comment Analogues of Primitive Recursive Functions
@Carlo It is relevant, but I am asking for something more refined: I want to consider not just which sets are $\Sigma_1$-definable in $A$, but when $A$ "knows" that one $\Sigma_1$-set is contained in another.
Jan
21
asked Analogues of Primitive Recursive Functions
Jan
21
revised $(\infty, 1)$-Yoneda embedding via the Grothendieck construction
deleted 1 character in body
Jan
21
revised Lower Algebra: Modules over the monoidal category of abelian groups
added 200 characters in body
Jan
21
awarded  Enlightened
Jan
21
awarded  Nice Answer