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For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

4
votes
0answers
116 views

What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

A paradox: Goodwillie calculus considers only finitary functors. $TC$ isn't finitary. Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem. (...
7
votes
1answer
139 views

Simplicial nerve functor commutes with opposites

There are two "opposite" functors: $$ op_\Delta\colon sSet\to sSet$$ and $$op_s\colon sCat\to sCat.$$ The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite ...
8
votes
0answers
92 views

Is every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?

Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For ...
6
votes
1answer
74 views

Is a weak functor which strictly preserves horizontal composition and which runs between strict bicategories automatically strict?

Let $\mathbf{B}$ and $\mathbf{B'}$ be strict bicategories and $F: \mathbf{B} \to \mathbf{B'}$ a weak functor which preserves horizontal composition strictly (i.e. $Ff * Fg = F(f * g)$ natural in f and ...
4
votes
0answers
82 views

Minimal infinity categories in the Segal space picture

There is a well-known notion of a minimal Kan complex (see Goerss/Jardine's book) which is generalized to a minimal quasi-category in these notes by Joyal www.math.uchicago.edu/~may/IMA/Joyal.pdf (...
9
votes
2answers
367 views

What are the reflective subcategories of the category of presentable categories?

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well. A nice property of presentable $\infty$-categories is that if ...
3
votes
1answer
98 views

2-morphisms for Bord(n)

I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, and have a (I guess) pretty basic question for my understanding of the (...
6
votes
1answer
199 views

Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity?

In homotopy theory, the word "norm" is commonly used in two different ways (well, surely there are other ways, but these two have a particular familial resemblence). Let $G$ be a finite group. A $G$-...
12
votes
1answer
252 views

What is the coskeleton tower of a quasi-category?

I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely,...
15
votes
2answers
290 views

How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor $$ d: \mathcal{C} \longrightarrow \mathcal{C}^{op} $$ such that $d(x)$ is dual to $x$ for ...
4
votes
0answers
270 views

Nonabelian Poincare duality

Nonabelian Poincare duality is introduced by Jacob Lurie in "Higher Algebra", Section 5.5.6. It seems, that my question is closely related to this definition. Question: what can one say about the ...
2
votes
1answer
66 views

Is the site for cubical sets with connections equivalent to a full subcategory of posets?

Cubical sets with connection form a presheaf category on some category $C$. Is $C$ just the full subcategory of the category of posets whose objects are products of the interval $\Delta[1]$?
3
votes
0answers
98 views

Which kind of functors preserve the bar-construction?

Let C, D be monoidal infinity categories that admit geometric realizations. Let $F: C \to D$ be a monoidal functor and A an augmented associative algebra of C. Denote $Bar(A)= \mathbb{1} \otimes_A \...
2
votes
1answer
90 views

Specifying complexes in quasicategories via squares

Let $J$ be an interval of integers viewed as a linearly ordered set, and let $I \subseteq \mathbf{N}(J)$ be the subsimplicial set given by the union of the elementary edges $(x, x+1)$. The inclusion $...
13
votes
0answers
206 views

Localizing $\mathrm{CombModCat}$ at the Quillen equivalences

Let $\mathrm{CombModCat}$ be the category of combinatorial model categories with left Quillen functors between them. By Dugger's theorem and the appendix of Lurie's "Higher Topos Theory" it ought to ...
3
votes
1answer
185 views

Definition of $(\infty,1)$-category in HoTT [duplicate]

Reading Lurie's Higher topos theory got me thinking, if we can think of $(\infty,1)-$categories as (1-)categories enriched over $\infty$-groupoid, then wouldn't the internal definition of an $(\infty,...
6
votes
2answers
260 views

Are cofibrations accessible?

The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations? More generally, let $C$ be a locally presentable ...
9
votes
0answers
154 views

Is there an “injective version” of the Bergner model structure?

The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions. ...
3
votes
0answers
75 views

What is the structure required to construct this homotopy of maps between mapping cones?

Let $X,Y$ be topological spaces, let $f$ be a continuous map from $X$ to $Y$ and let $g$ be a continuous map from $Y$ to $X$. Write $C_f$ for the mapping cone of $f$; i.e., $\{*\} + X\times I + Y$, ...
4
votes
0answers
65 views

Is there a good, general description of morphisms right orthogonal to effective epimorphisms?

Let $C$ be a locally presentable, locally cartesian closed $\infty$-category. Then I think it's not hard to show that the class of effective epimorphisms in $C$ is closed under colimits and cobase-...
5
votes
0answers
115 views

Applications of spectral Artin representability?

The spectral Artin representability theorem says that a functor $X:CAlg^{cn}\rightarrow S$ from the $\infty$-category of connective $E_{\infty}$-rings to the $\infty$-category of small topological ...
4
votes
0answers
66 views

Internal van Kampen colimits

Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$. We can $C$-internalize everything in sight: Let $\...
18
votes
1answer
621 views

Spectral algebraic geometry vs derived algebraic geometry in positive characteristic?

Let $R$ be a commutative ring. Then there is a forgetful functor from the $\infty$-category of simplicial commutative $R$-algebras to the $\infty$-category of connective $E_{\infty}$-algebras over $R$....
7
votes
1answer
209 views

What is the relationship between $O_G$-parameterized $\infty$-categories and $\infty$-categories enriched in $Top_G$?

Let $G$ be a finite group. Barwick et al define a $G-\infty$-category to be a fibration over the orbit category $O_G$ of transitive $G$-sets. But in the non-$\infty$-land, the natural guess at where I ...
3
votes
0answers
52 views

Calculating the intersection of the saturations of a decreasing sequence of morphisms

I have a sequence $W_0\supset W_1\supset \ldots$ of classes of morphisms in a presentable $(\infty,1)$-category $\mathcal{M}$. I can present this $(\infty,1)$-category as a model category, but I'm ...
2
votes
2answers
181 views

Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data

Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step ...
13
votes
2answers
578 views

What kind of category is generated by Cubical type theory?

What kind of ‘category’ is Cubical type theory the internal language of? Its known that Martin-Löf type theories are the internal language of Locally cartesian closed categories, adding higher ...
5
votes
1answer
96 views

Left split subobject in a $2$-category

Let $\mathcal{K}$ be a $2$-category. Keeping in mind the Cat-intuition on $\mathcal{K}$ I say that: Def : A $1$-cell $A \stackrel{f}{\to} B$ is a left split subobject if $f$ is a right adjoint and ...
5
votes
1answer
129 views

Explicit expression of the unstraightening functor

Hard as I tried, I couldn't find a proof of Remark 2.2.2.11 in Higher Topos Theory, or prove it myself. It seems to need an explicit formulation for the unstraightening functor, so my question is: is ...
9
votes
2answers
448 views

2-natural operations on toposes

Any pseudonatural endomorphism $\Phi$ of the forgetful 2-functor $U:Topos^{coop}\to Cat$ is essentially determined by its component $\Phi_{Set}$. But which endofunctors of $Set$ induce such a $\Phi$? ...
14
votes
3answers
478 views

Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categories?

Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences. Let $\mathbf Q$ be the corresponding $\infty$-...
8
votes
1answer
145 views

Weak colimits in locally cartesian closed categories

The general adjoint functor theorem implies that a complete locally small category has a weak colimit of a diagram if and only if it has a colimit of this diagram. It seems that this is also true for ...
13
votes
1answer
274 views

Weak complicial sets: Are the morphisms too strict?

In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified ...
13
votes
2answers
417 views

What is a model category from an $\infty$ point of view?

A number of different models for $\infty$ categories can seen to have analogs in $\infty$-category theory. For example: Quasicategories: $\Delta \subseteq \mathrm{Cat}_{(\infty, 1)}$ is (?) a dense ...
8
votes
0answers
188 views

How to define the direct sum of TQFTs $(\infty,1)$-categorically?

Let $\mathit{Bord}_d$ be the symmetric monoidal category of $(d-1)$-manifolds and bordisms between them. Let $\mathcal{C}$ be the symmetric monoidal category of $k$-modules. Then, for two symmetric ...
8
votes
1answer
155 views

Gurski's Definition of a strict functor of tricategories

In Gurskis definition (page 32 of his thesis) of a strict functor $F$ of tricategories he requires that $F$ maps the adjoint equivalences $a,l,r$ in the source tricategory to the same adjoint ...
7
votes
1answer
150 views

Can an enriched functor be expressed as a colimit of representable functors?

Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors ...
6
votes
1answer
119 views

Is there a general way to turn a 2-monad into a lax-idempotent (a.k.a. KZ) 2-monad?

Often a 2-monad is best replaced with a KZ monad. For example: $Fun(B,Cat)$ is 2-monadic over $Cat/Ob B$, but KZ over $Cat/B$. $SymMonCat$ is 2-monadic over $Cat$, but KZ over $Cat/Fin_\ast$. The ...
10
votes
0answers
141 views

Is every colimit-generator dense in an $\infty$-topos?

Recall that there are various senses in which a full subcategory $G \subseteq C$ may "generate" a category $C$. For example, in order of increasing strength (under reasonable conditions): $G$ is a ...
4
votes
1answer
300 views

Kan condition for bar construction

Let $T$ be a monad on a concrete category $\mathcal{C}$, and $A$ an algebra over $T$. The bar construction is a simplicial object in the category $\mathcal{C}^T$ of algebras which we can think of a ...
11
votes
2answers
1k views

Questions about categorification (& combinatorial simplification of the Russian approach to Lusztig's conjectures, in zero & positive characteristic) [closed]

The goal of to give a uniform approach to the Russian proofs of Lusztig's conjectures using higher representation theory (extending [Elias-Williamson]'s work on the Kazhdan-Lusztig conjecture). ...
2
votes
1answer
86 views

When is the derived category $D(A)$ locally cartesian closed?

Let $D(A)$ be the derived $(\infty,1)$-category of some abelian category $A$. For which $A$ is $D(A)$ locally cartesian closed? Replace $D$ with $D^b$ or similar if appropriate. I essentially want ...
8
votes
1answer
265 views

Example of a (presentable $k$-linear $\infty$-)category which is dualizable but not compactly generated?

Is there an example of a presentable, stable, $k$-linear $\infty$-category which is dualizable but not compactly generated, where $k$ has characteristic zero, and which is $\text{QCoh}(X)$ (by which I ...
6
votes
0answers
130 views

Double-categorical refinement of twisted arrow category: does it have a name?

Let $C$ be a category. The twisted arrow category $Tw(C)$ can be refined to a double category $TTw(C)$ by making morphisms on the left "vertical" and morphisms on the right "horizontal". Question: I'...
5
votes
2answers
182 views

Does the Day convolution induce the structure of a bimonoidal category on $Fun(C,D)$?

Let $\mathcal{C}$ and $\mathcal{D}$ be symmetric monoidal categories and assume that the symmetric monoidal product $\otimes_{\mathcal{D}}$ on $\mathcal{D}$ preserves colimits in both variables. Then ...
11
votes
2answers
284 views

$(\infty,2)$-Categorical Analogue of the Local Nature of Equivalences

It is well known that, for two functors $F,G : I \to C$ for $I,C$ some $\infty$-categories, the property that a map $\phi: F \to G$ is an equivalence can be checked locally on $I$. Namely, if $\phi(i) ...
15
votes
1answer
541 views

A sheaf is a presheaf that preserves small limits

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough. However when reading ...
4
votes
1answer
159 views

Classes of monomorphisms - a definition

I have a certain construction relating to subobjects in an arbitrary category. Now the nlab article on subobjects says: More generally, in some contexts we may take “subobject” to mean an ...
3
votes
1answer
141 views

When the global section functor is a Cartesian fibration?

Given a Cartesian fibration $p : \mathbf{E} \to \mathbf{B}$ over an $\infty$-topos the paper by Marc Hoyois mentioned in his answer to this question gives some sufficient conditions for $\mathbf{E}$ ...
4
votes
2answers
160 views

Generalizations of tangent $\infty$-topos

If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\...