**10**

votes

**2**answers

276 views

### Is this a functor on the category of $C^{*}$ algebras?

The category of $C^{*}$ algebras is denoted by $\mathcal{A}$.
Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, ...

**7**

votes

**1**answer

320 views

### How (if at all) does category theory deal with situations where the usual notion of isomorphism isn't right?

This question will potentially rub some people the wrong way; I can't do much about this, except state right here at the outset that this question is motivated by a genuine desire to understand, and ...

**6**

votes

**2**answers

329 views

### Is dgCat a category or a 2-category?

Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has ...

**11**

votes

**1**answer

297 views

### Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question.
In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...

**10**

votes

**1**answer

200 views

### Relationship between two universal properties of the category of elements?

Let $G: \mathcal{A} \to \mathsf{Set}$ be a functor. Recall that the category of elements $\mathsf{el}G$ is defined by the comma square
$\require{AMScd}$
\begin{CD}
\mathsf{el}G @>!>> ...

**3**

votes

**1**answer

162 views

### Universal covering and double cover functors

Initially posted on MSE
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...

**13**

votes

**1**answer

217 views

### Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds

Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming ...

**7**

votes

**1**answer

97 views

### Characterisations of closed embeddings in $Top_1$?

Let $Top_1$ be the category of topological spaces which are $T_1.$
I am curious as to whether there is a categorical definition of what a closed embedding is in this environment. With a ...

**8**

votes

**2**answers

296 views

### The category of elements, enrichment, and weighted limits

This is a crosspost of this MSE question.
Every so often, when reading notes online or skimming through books, the category of elements and the Grothendieck construction pop up. I don't know ...

**6**

votes

**1**answer

360 views

### Iterated Homotopy Quotient

If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can ...

**8**

votes

**0**answers

69 views

### W-types and inverse image functor

All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.
I would like to know whether the ...

**5**

votes

**1**answer

122 views

### For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?

I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at ...

**2**

votes

**0**answers

107 views

### Sufficient criteria for a nerve of a topological category to be good

I know that the following statement is true and I am looking for a reference:
Given a topological category $\mathcal C$ (i.e. morphisms and objects form a space and all maps in the definition of a ...

**2**

votes

**0**answers

86 views

### About the definition of lax.functor between tricategories

SUMMARY: Observing that monoids in a monoidal category are identified with lax.functors (with domain 1), I tried to generalize this argument wanting to get a skew-Monoidal-category as ...

**5**

votes

**1**answer

194 views

### Minimal model (resolution) for a specific colored operad

We know that for the operad $As:=\mathcal{F}(\mu)/(\mu\circ_1\mu-\mu\circ_2\mu)$, its minimal model is the free operad $\mathcal{F}(E)$ where $E=\mathbb{k}<\mu_2,\mu_3,\dots,\mu_n,\dots>$ is the ...

**2**

votes

**0**answers

49 views

### Cancellation property of groupoidal cartesian fibrations

I have an issue concerning a property of "left cancellation" for groupoidal cartesian fibrations of ∞-cosmoi (but everything works fine in a 2-category as well).
A 1-cell $p: E \to B$ is called ...

**8**

votes

**1**answer

218 views

### Is every locally compactly generated space compactly generated?

[Parse it as (locally compact)ly generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...

**26**

votes

**2**answers

955 views

### Why is there a duality between spaces and commutative algebras?

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated algebras over $\mathbb{C}$. The equivalence associates to an affine variety its ...

**4**

votes

**1**answer

127 views

### Enriched Cauchy completions and underlying categories

The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] ...

**3**

votes

**0**answers

117 views

### Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety

Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$.
I read ...

**7**

votes

**0**answers

166 views

### 6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ ...

**5**

votes

**0**answers

64 views

### Does the $D$-property have universal objects?

A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...

**2**

votes

**0**answers

183 views

### Can such categorical notion of action be formalized?

I'm wondering if it's possible to find an universal construction for a general concept of action for (single-sorted?) finite product sketches, such that one of those is "acting" on the second in the ...

**8**

votes

**3**answers

879 views

### Category theory for Algebraic Geometry

How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?

**1**

vote

**0**answers

95 views

### Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...

**5**

votes

**1**answer

137 views

### Kan extensions of pseudofunctors

Can anyone suggest a reference for (left) Kan extensions of pseudofunctors?
In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr ...

**16**

votes

**3**answers

420 views

### Brouwer's theorem for the Cauchy reals

Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...

**18**

votes

**0**answers

578 views

### A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that ...

**4**

votes

**1**answer

267 views

### The “$\infty$”-column in the periodic table of n-categories

A monoid is the same as a category with a single object.
A monoidal category is the same as a bi-category with a single object.
A commutative monoid is the same as a bi-category with a single object ...

**12**

votes

**2**answers

403 views

### Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction.
Given diagrams of topological spaces
$$X_0\rightarrow X_1\rightarrow\ldots$$
$$Y_0\rightarrow Y_1\rightarrow\ldots$$
...

**3**

votes

**0**answers

76 views

### Natural transformations of $A_\infty$-functors (between dg-categories) are “directed homotopies” (reference?)

Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to ...

**3**

votes

**0**answers

97 views

### Does this notion of “$\mathcal{F}$-digraph” appear in the literature?

By a digraph, I mean a quiver with no multiple edges. So in particular:
Loops are okay.
An infinite set of vertexes is okay.
Furthermore, I will tend to identify each digraph with its underlying ...

**4**

votes

**2**answers

295 views

### Algebras for probability monad

What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by
$$
DX = \left\{ p \in [0,1]^X \ ...

**-1**

votes

**1**answer

101 views

### Categorical product of graphs and chromatic number

Let $(G_i)_{i\in I}$ denote a family of simple, undirected graphs (finite or infinite). Let $\prod_{i\in I}G_i$ denote their categorical product. Why do we have the inequality
$$\chi(\prod_{i\in ...

**2**

votes

**0**answers

100 views

### About a (new?) definition of transformation (anti.transformation) as a link between natural and dinatural transformations

This is not a hard topic, but I post here as "reference request" or because elementary aspects (but not previously vocalized) can be interesting too for researchers.
Given $F: \mathscr{A}\to ...

**2**

votes

**1**answer

95 views

### Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors
$F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the ...

**10**

votes

**2**answers

650 views

### What's a good introduction to category theory for someone doing analysis?

I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory.
Any recommendations?

**2**

votes

**0**answers

98 views

### Generalizing disjointness

The following definition generalizes set-theoretic disjointess:
Definition 0. (Autonomy). Given a Lawvere theory $\mathsf{T}$, a $\mathsf{T}$-algebra $X$, and an indexed family $S$ of subalgebras ...

**1**

vote

**1**answer

100 views

### How to define the internal hom between presheaves valued in cotensored categories?

First let $\mathcal{V}$ be a closed symmetric monoidal category
and $\mathcal{M}$ be a category enriched over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is cotensored, or powered over ...

**4**

votes

**0**answers

178 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

62 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 ...

**10**

votes

**1**answer

322 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 ...

**35**

votes

**2**answers

938 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 ...

**16**

votes

**1**answer

372 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

150 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

53 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 $\mathscr{D}$ ...

**4**

votes

**2**answers

298 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

378 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

104 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

177 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 ...