**4**

votes

**0**answers

161 views

### Schwede-Shipley theorem for monoidal categories?

The Schwede-Shipley theorem gives a criterion for a presentable stable $\infty$-category to be the category of modules over an $\mathcal{E}_1$-algebra. Is there any similar criterion for a monoidal ...

**0**

votes

**0**answers

57 views

### A construction on lax.functor

Consider for simplicity only locally small 2-categories.
Given a 2-category $\mathscr{A}$ let $|\mathscr{A}|$ its 2-graph (forget the horizontal composition).
Given a 2-graph $\mathcal{G}$ let ...

**9**

votes

**1**answer

283 views

### Adding inverses to a symmetric monoidal category (Reference?)

As we all know, the forgetful functor $\mathsf{Ab} \to \mathsf{CMon}$ from abelian groups to commutative monoids has a left adjoint, the Grothendieck group. I would like to categorify this ...

**34**

votes

**2**answers

864 views

### The formal p-adic numbers

The real numbers can be defined in two ways (well, more than two, but let's stick to these for now): as the Cauchy completion of the metric space $\mathbb{Q}$ with its usual absolute value, or as the ...

**15**

votes

**1**answer

343 views

### The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...

**5**

votes

**1**answer

147 views

### If $C$ has all geometric realizations of simplicial objects, what other colimits does it have?

Let $C$ be an $\infty$-category. Suppose that every diagram $\Delta^{\mathit{op}} \to C$ has a colimit. Is there any characterization of small categories $I$ such that every diagram $I \to C$ has a ...

**0**

votes

**0**answers

50 views

### About the functors composition completeness

If $F: \mathscr{A}\to \mathscr{B}$ is a functor and $[\mathscr{C}, \mathscr{D}]$ is the category of functors and natural transformations between two given categories $\mathscr{C}$ and $\mathscr{D}$ ...

**1**

vote

**0**answers

120 views

### History of categorical localization sans calculi of fractions

This question arises from a paper which I've just found and skimmed:
FW Bauer, J Dugundji. Categorical homotopy and fibrations. Transactions of the American Mathematical Society, 1969
With 28 ...

**4**

votes

**2**answers

287 views

### Categories of finite objects

In my experience, category theory is very successful at providing powerful machinery to reason about large objects or objects unrestricted in size, for example (logical) models (via accessible ...

**2**

votes

**2**answers

355 views

### Reference for higher categorical analogue of algebraic cycle? [closed]

Are there higher categorical analogues of algebraic cycles?
What are some references?
This question arise in an attempt to generalize algebraic cycles towards higher dimensional algebra. Has there ...

**1**

vote

**1**answer

100 views

### Can hypercompletion be an essential localization?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory, i.e. its inclusion functor has a left-exact left ...

**4**

votes

**1**answer

149 views

### locally noetherian categories and the category of quasi-coherent sheaves over a noetherian scheme

It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules
are injective if and only if every injective $R$-module is a direct sum of indecomposable injective ...

**6**

votes

**1**answer

284 views

### Can hypercomplete objects be coreflective?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory. Can it ever happen that this subcategory is ...

**3**

votes

**1**answer

68 views

### Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects.
Let $({\cal C},\otimes,*)$ be a semisimple ...

**3**

votes

**1**answer

148 views

### Is a pullback along a Dold fibration a homotopy pullback?

Let $$
\begin{array}{ccc}
A & \to & B
\cr\downarrow&&\downarrow
\cr
A'& \to &B'
\end{array}
$$ be a pullback square in the category of all topological spaces (not just in a ...

**3**

votes

**1**answer

116 views

### About a closed strucure on profunctors

Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$.
...

**8**

votes

**1**answer

269 views

### Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...

**2**

votes

**1**answer

99 views

### Do the full subcategories have a simple structure in higher category theory?

Let $C \in Cat$ be an $(\infty,1)$-category.
Let $P$ be the partially ordered subset of full subcategories of $C$.
Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer ...

**8**

votes

**1**answer

122 views

### Detecting positive endomaps of the formal reals

A locale is a sort of "formal topological space", which "may not have enough points to separate its open sets". For instance, there is a "locale of all real numbers that are both rational and ...

**0**

votes

**1**answer

105 views

### Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?

**2**

votes

**1**answer

106 views

### Does the following characterize local presentability?

Let $\mathcal C$ be a cocomplete category. Consider the following two conditions:
$\mathcal C$ is locally presentable.
The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors ...

**10**

votes

**1**answer

204 views

### Does every Lawvere theory arise in this way?

By a Lawvere theory, I mean a finite-product category $\mathsf{T}$ equipped with a distinguished object, such that every object of $\mathsf{T}$ can be expressed as a finite product of the ...

**4**

votes

**1**answer

188 views

### An example for a construction on monads/operads?

Suppose that $C$ is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A ...

**5**

votes

**0**answers

147 views

### “Generalized theory of polynomials” for a given commutative Lawvere Theory

I am trying to understand
Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing.
Let $R$ be a commutative, associative ring with unit. We can ...

**2**

votes

**2**answers

199 views

### When is the adjoint to a monoidal functor monoidal?

Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is lax monoidal if it is equipped with maps $1_{\mathcal D} \to F(1_{\mathcal C})$ and $F(X) ...

**3**

votes

**0**answers

103 views

### What about “bilax” functors?

in [G] p.29, J.W Gray define the 2-comma category $[F, G]$ of two 2-functors $F: \mathcal{A}\to \mathcal{D},\ G: \mathcal{B}\to \mathcal{D}$. This definition work well also if we suppose $F$ a ...

**3**

votes

**0**answers

235 views

### Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...

**2**

votes

**1**answer

99 views

### Exponential locales and a pointless version of the compact-open topology?

TL;DR: compact-open topology for Homs of locales?
Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.
For two locales, $A$ and $B$, is there a nice way to make an ...

**0**

votes

**1**answer

96 views

### Yetter-Drinfeld modules as rigid category

I'm reading a proof of the following theorem
If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category.
In this proof we define ...

**6**

votes

**1**answer

202 views

### When the restriction of a derived functor to a subcategory is the derived functor of the restriction

Let $\mathcal{D},\mathcal{E}$ be abelian categories and $\mathcal{C}$ be a Serre subcategory of $\mathcal{D}$.
Let $D(\mathcal{C}), \, D(\mathcal{D})$ denote the derived categories of ...

**4**

votes

**1**answer

210 views

### Properties of loop space functor from homotopy types to group objects in homotopy types

I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important:
When we take the loop-space of a (connected) homotopy type, we get ...

**1**

vote

**2**answers

156 views

### Completion under weighted limits/colimits

Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance

**3**

votes

**0**answers

179 views

### Ring epimorphisms, and epimorphism in the category of small preadditive cats

This question is related to this other question I have asked some time ago. Let $R$ and $S$ be two rings and let $\phi:R\to S$ be a ring homomorphism.
It is well-known that $\phi$ is an epimorphism ...

**10**

votes

**2**answers

403 views

### Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

I asked this some days ago over at math.se, and while the question got 10 upvotes, I didn't get too many answers. Although it is a "soft question", maybe the general issue is interesting enough to ...

**1**

vote

**0**answers

46 views

### About cylinder and path functors

Let $\mathcal{K}$ be a locally presentable category. I recall that a cylinder $C:\mathcal{K}\to \mathcal{K}$ is by definition equipped with two natural maps $\gamma_X:X\sqcup X\to CX$ and ...

**11**

votes

**2**answers

491 views

### Is there a “free abelian group of rank 1” in the category of affine group schemes?

Let's fix an algebraically closed field $k$.
The group $\mathbb Z$, as a discrete group scheme, is not affine since it's not quasi-compact. Is there an affine algebraic scheme over $k$ whose ...

**5**

votes

**1**answer

154 views

### Do models-and-homomorphisms always form an accessible category?

It's well-known that the category of models of any first-order theory $T$ form an accessible category if we take the elementary embeddings as morphisms. This is true in finitary first-order logic or ...

**4**

votes

**1**answer

283 views

### Natural transformations induce homotopies - Is this true in the “fat” world?

Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors ...

**4**

votes

**0**answers

170 views

### Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...

**2**

votes

**1**answer

173 views

### Why do we need filtered categories to index ind-objects?

I edited the question in view of several helpful replies (thanks).
When we define ind-objects in a category, we use in general filtered diagrams in a category, not just sequences $A_1 \rightarrow ...

**1**

vote

**1**answer

168 views

### “Canonical” graph structure on $\text{Hom}(G, H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...

**12**

votes

**1**answer

447 views

### Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...

**1**

vote

**0**answers

65 views

### Reduced products of (abelian and triangulated) categories: references?

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...

**0**

votes

**0**answers

122 views

### Higher algebra and terminology about 2-objects

It is well known that one way to build higher category theory is to use some induction process, where an $n$-category has as $0$-cells some $n-1$ categories, such that for two $0$-cells $\mathcal{A}$ ...

**7**

votes

**2**answers

532 views

### What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a ...

**1**

vote

**1**answer

140 views

### Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?

Marc's answer to my previous question gives a way to compute colimits in the category of presentable $\infty$-categories and continuous functors, using the (discontinuous) right adjoints to those ...

**3**

votes

**1**answer

127 views

### What is the cokernel of a map of presentable stable $\infty$-categories?

Let $C$ and $D$ be presentable stable $\infty$-categories, and let $f:C \to D$ be a continuous functor between them. Let $0$ be the trivial stable $\infty$-category. What is the colimit of the ...

**5**

votes

**2**answers

320 views

### Is the $\infty$-category of presentable $\infty$-categories presentable?

Let $\mathit{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and continuous functors in some universe. Is it presentable itself a larger universe?

**13**

votes

**2**answers

340 views

### Rectifying the definition of a closed category

The definition of a closed category I'm using is here.
Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal ...

**0**

votes

**0**answers

315 views

### Set as a (strict) infinite-category?

First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold:
1) trying to ...