**15**

votes

**1**answer

325 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

139 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

119 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

283 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

343 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

95 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

141 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

271 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

62 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

144 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

112 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

255 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

96 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

115 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

100 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

103 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

191 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

178 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

142 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

181 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

100 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

232 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

98 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

90 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

195 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

204 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

154 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

172 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 ...

**7**

votes

**1**answer

301 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

42 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

479 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

148 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

274 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

156 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

171 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

167 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

442 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

119 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

507 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

135 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

120 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

313 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

331 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

310 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 ...

**1**

vote

**0**answers

61 views

### Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"
In Part (2) of Theorem 19.8.4 of that book it says
If ...

**2**

votes

**0**answers

63 views

### TTF triples are recollements

The notion of recollement
$$
\mathcal{A}'
\stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}}
...

**1**

vote

**0**answers

84 views

### $n$-recollements and perverse t-structures

A recent preprint on arXiv brought my attention on the notion of $n$-recollement (def. 2) a generalization of the notion of recollement among three abelian or triangulated categories behaving like a ...

**1**

vote

**0**answers

159 views

### Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...