# Questions tagged [higher-category-theory]

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

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### 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

**0**answers

92 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
(...

**11**

votes

**2**answers

433 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

**1**answer

114 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

**1**answer

264 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 $G$-...

**13**

votes

**1**answer

279 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**

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310 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

**0**answers

312 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

**1**answer

72 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

**0**answers

115 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

**1**answer

95 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

**0**answers

222 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

**1**answer

209 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,...

**7**

votes

**2**answers

279 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 ...

**10**

votes

**0**answers

177 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

**0**answers

78 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

**0**answers

70 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

**0**answers

136 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 ...

**6**

votes

**0**answers

76 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

**1**answer

860 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

**1**answer

222 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**

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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

**2**answers

182 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

**2**answers

657 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

**1**answer

106 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

**1**answer

156 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

**2**answers

465 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

**3**answers

522 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

**1**answer

153 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

**1**answer

289 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

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443 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**

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198 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

**1**answer

159 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

**1**answer

164 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

**1**answer

131 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 ...

**11**

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152 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

**1**answer

306 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 ...

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1k views

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

This is a question about the proofs of Kazhdan-Lusztig's conjectures for category $\mathcal{O}$ using higher representation theory (avoiding Beilinson-Bernstein's geometric localization theory).
...

**2**

votes

**1**answer

91 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

**1**answer

308 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

**0**answers

132 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'...

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votes

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188 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

**2**answers

292 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) ...

**16**

votes

**1**answer

586 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

**1**answer

165 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

**1**answer

155 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 $\mathbf{E}$ ...

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votes

**2**answers

173 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 $\...

**7**

votes

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425 views

### Are categories of fibrant objects idempotent complete?

If $C$ is a category of fibrant objects, is its associated $\infty$-category idempotent complete, i.e. is it accessible? If this is not always true, besides from the case when it is an $n$-category ...

**5**

votes

**0**answers

196 views

### Cofree conilpotent (cocommutative) coalgebra for $\infty$-categories

Let $\mathrm{K}$ be a field.
Denote $ \mathrm{Vect}_{\mathrm{K}} $ the category of K-vector spaces, $\mathrm{Coalg }^{\mathrm{conil } } $ the category of
conilpotent, coaugmented, coassociative ...

**10**

votes

**1**answer

291 views

### Gabriel-Ulmer duality for $\infty$-categories

Gabriel-Ulmer duality states that 2-categories $\mathrm{Lex}$ (of small finitely complete categories and functors preserving finite limits) and $\mathrm{LFP}$ (of locally finitely presentable ...