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

6
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0answers
101 views

What is a spectrum object in $\infty$-topoi?

For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra? To ...
3
votes
0answers
108 views

Is $Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))$ for an additive category $\mathcal{C}$?

Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category. ...
9
votes
0answers
202 views

What are the monomorphisms of ($\infty$-)toposes?

There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...
10
votes
1answer
240 views

Are there continua in $\infty$-topoi?

If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence....
3
votes
1answer
208 views

Why must the essential image break the principle of equivalence?

I'm having trouble understanding why the "essential image" is defined the way it is. The nlab article gives the following definition: (A concrete realization of) the essential image of a functor $...
9
votes
0answers
197 views

Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures

I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one. Suppose we are given a (strict) pullback square ...
8
votes
1answer
387 views

Spectral and derived deformations of schemes

I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is. Let $S = (X, ...
34
votes
2answers
743 views

What parts of the theory of quasicategories have been simplified since the publication of HTT?

It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression ...
7
votes
4answers
370 views

Localization of $\infty$-categories

In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...
3
votes
0answers
174 views

Cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$

Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group. (For example, given a short exact sequence $$ 1 \to BG_2 \to \mathbb{G} \to G_1 \to 1 $$ and the fiber sequence: $$ B^2G_2 ...
2
votes
1answer
150 views

Relaxing a natural isomorphism to a natural transformation to obtain a more general $2$-category

Many definitions of $2$-categories are given as categories equipped with some extra structure encoded by some functors and some natural (or extranatural) transformations between these functors (or ...
7
votes
1answer
152 views

Direct comparison from the Rezk hom to the hom of a simplicial category along the coherent nerve?

Consider the following construction: Define $G_n$ to be the contractible groupoid on $n+1$ objects. Choosing a linear order on the objects of each $G_n$ turns $G_*$ into a cosimplicial object. ...
3
votes
0answers
136 views

A naive question about representations of group stacks

For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with ...
8
votes
0answers
199 views

In what context can enriched category theory be done?

There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list. My question is what ...
6
votes
1answer
261 views

Physical consequences of cobordism hypothesis?

Let $C$ be a symmetric monoidal $n$-category. An extended framed $C$-valued TQFT is a symmetric monoidal functor from the framed bordism category $\mathrm{Cob}^{fr}_n(n)$ to $C$. The cobordism ...
10
votes
0answers
203 views

Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack

I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8. ...
7
votes
1answer
189 views

Monoidal structures on modules over derived coalgebras

Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can ...
4
votes
1answer
132 views

What is the idempotent completion of the (2,1)-category of spans of finite sets?

I don't believe the $(2,1)$-category $FinSpan$ has split idempotents. Question: Is there a simple description of the idempotent completion of $FinSpan$? Foundationally, we may think of $FinSpan$ as ...
5
votes
0answers
92 views

Internal hom as 2-Kan lift of pseudofunctor

Consider a situation where there is a pseudofunctor from some category $C$. Due to it's geometrical properties, it might induce some sort of an internal hom on the original category (making it a ...
7
votes
2answers
332 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
155 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
99 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
78 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
83 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
385 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 ...
4
votes
1answer
109 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
218 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
267 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
303 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
289 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
69 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
110 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
94 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
217 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
200 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
274 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
166 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
77 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
69 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
123 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
69 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 $\...
19
votes
1answer
720 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
220 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
54 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
624 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
101 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 ...
6
votes
1answer
135 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
452 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 $\Phi$? ...
15
votes
3answers
500 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 $\infty$-...