Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,313
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Model structure on stacks
does anyone know if there is a model structure on stacks where the cofibrations are the monomorphisms ?
As far as I know usually to get a model structure on stacks one localizes a model structure on ...
user2664
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What is a Homotopy between $L_\infty$-algebra morphisms II
I would like to continue on question I asking, what is a homotopy between
Lie infinity algebras, since I'm not satisfied in two directions:
1.) The naive approach to define a homotopy would be ('...
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2 and 3 pullbacks
If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times_C C^I \times_C B$, where $I$ is the ...
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Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology
Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this $(...
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Higher dimensional pasting diagram of cubes
A pasting diagram is the analogue of composition of arrows in higher categories. The nlab page gives an example of a two dimensional diagram and how to compute it as a composite of two morphisms. I am ...
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What is the suitable setting for supercoherence with value in a bicategory?
It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as ...
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Are topoi and etale geometric morphisms locally small?
This question is related to this one: Local smallness and (higher) topoi which has not yet been answered.
The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in ...
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Do Homotopy Fully Faithful Functors Push-out?
A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.
Are homotopy fully faithful functors preserved under (homotopy) pushout?
More ...
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Joins of, and limits in, $(\infty,1)$-categories via profunctors
I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind.
In the classical setting it is almost a triviality to express the join ...
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Local smallness and (higher) topoi
The $2$-category of topoi and geometric morphisms is not locally small. For example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is ...
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How do you define (infinity,1) categories in Homotopy Type Theory?
One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...
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Create 2-category from a cartesian category
I have a way of constructing (something like) a 2-category from a category with products $\mathcal{C}$.
My question: Is this a correct construction, and if so, does it have a name?
Define $\mathcal{...
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Is there a Rado category?
The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between finite/...
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Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?
Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
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What interesting homotopy invariants can I write down using the universal property of homotopy types?
I've recently been led to believe some version of the following statement:
Weak homotopy types, or equivalently $\infty$-groupoids (let me not commit myself to a particular model of these), are ...
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What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?
Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence ...
- 8,707
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What's the link between topological spaces as locales and topological spaces as infinity-groupoids?
I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...
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About a Double-pseudo-category generalization of the module bicategory construction
To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is ...
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Loops and suspensions of higher categories
Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ ...
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Lower Algebra: Modules over the monoidal category of abelian groups
Proposition 6.3.2.18 of Higher Algebra identifies $Mod_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable ...
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The state of the art in the rectification of homotopy-coherent structures
My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of:
Cordier and Porter proved a ...
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For an $(\infty,1)$-topos, is the object functor from groupoid objects to the topos a fibration, cofibration?
This question is kind of suggested by the question Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?
Question: If $\mathcal C$ in an $(\...
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Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?
Let $C$ be an $(\infty,1)$-topos. The $(\infty,1)$-category of group objects in $C$ is a full sub-$(\infty,1)$-category of groupoid objects in $C$:
$${\mathsf{Grp}}(C) \hookrightarrow {\mathsf{Grpd}}(...
user2529
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Is there a Wall finiteness obstruction in other settings?
Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW ...
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categorical characterization of large cardinals
Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...
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In the topos-theoretic interpretation of Physics by Isham & Doering what role does intuitionistic logic play? [closed]
I've asked this question on Physics.SE but was advised to ask it here.
Isham & Doering have written a series of papers exploring how to ground physics in topoi. Now the internal logic of topoi is ...
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Compact objects in undercategories and filtered colimits
Let $\mathcal{C}$ be a compactly generated presentable $(\infty, 1)$-category. Consider the functor
$$ \Phi: \mathcal{C} \to \mathrm{Cat}_\infty, \quad x \mapsto (\mathcal{C}_{x/})^\omega,$$
that ...
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Categories of sheaves and Kan Extensions
This is quite a broad question regarding constructions of categories of sheaves in geometry.
Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...
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What kind of category is a cyclically ordered set?
Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
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(∞,n)-category of triangulated cobordisms
What is an accepted definition of a (∞,n)-category of triangulated cobordisms?
Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
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Homotopy-theoretic derived Morita equivalences
Recall that two $k$-algebras $A, B$ are Morita equivalent iff their categories of left modules are equivalent. However, this relation turns out to be rather fine and one introduces a coarser ...
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How does one Segal-subdivide a 2-category?
Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...
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Localisation in a quasi-category
Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction.
Now consider ...
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Is there an accepted definition of $(\infty,\infty)$ category?
For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
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Is Lemma A.1.5.7 in Higher Topos Theory correct?
Hello to everyone,
I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in section A.2.6.
At some ...
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how to make the category of chain complexes into an $(\infty,1)$-category
Related to this question, I would like to know, if there is an explicit presentation
of the $(\infty,1)$-category of a model category of abelian chain complexes, but this time in terms of simplicial ...
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KK-theory by abelianized correspondences of smooth stacks?
Whith (Kasparov, bivariant) KK-theory I am left with the nagging feeling that the theory is "more fundamental", than has been made explicit, that there is a "more profound" universal characterization ...
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Is the site of (smooth) manifolds hypercomplete?
By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
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Diagram spectra and Algebraic Geometry
I was recently reading a paper titled "Model Categories of Diagram Spectra" and it was mentioned in the paper that the contents of the paper were also useful in algebraic geometry. I'm really ...
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What are some examples of weak ω-categories?
As is usual, let's say an (n, k)-category is something with
objects, morphisms, 2-morphisms, ..., n-morphisms, such that all
j-morphisms for j > k are invertible, everything meant in the
weak sense. ...
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Infinity-categories vs Kan complexes
It is known (cf. Lurie's book Higher Topos Theory for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan complexes, aka ...
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What is the Q-construction, metaphysically?
An exact (small) category $P$ is an environment in which we make sense of the "put-together"-edness of objects via (short) exact sequences. It seems like the K-theory of an exact category encodes the ...
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looping and delooping spaces and categories
I'm trying to understand the relationship between the notions of looping and delooping in category theory and topology.
The morphisms in a category with one object have the structure of a monoid. ...
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What is a higher derived constructible sheaf
Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C_*(\Omega X,k)$ of ...
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Generalized categories for "higher homotopy groupoids"
I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...
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Bisections in Kan Complexes
Kan Complexes can be seen as a generalization of groupoids, mostly called (weak)
infinity groupoids in this context.
On groupoids we can define the \textbf{group of bisections} the following way:
...
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The category theory of $(\infty, 1)$-categories
There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...
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Are there non-categorical notions in topos theory?
Suppose that $\mathcal{T}$ is an abstract $2$-category we know is equivalent to the $2$-category of Grothendieck topoi via some equivalence $$\phi:\mathcal{T} \to \mathfrak{Top},$$ and let $E$ be an ...
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Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?
This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...
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Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?
The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO).
On the other hand, MU and ...
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