# Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

5,138
questions

**11**

votes

**0**answers

151 views

### What are internal complete atomic boolean algebras, intuitively?

The category of complete atomic boolean algebras $\mathbf{CABA}$ is equivalent to $\mathbf{Set}^{\mathrm{op}}$ via
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
...

**5**

votes

**1**answer

383 views

### Name for a set of elements that fully determine a morphism

In a concrete category (i.e., where the morphisms are functions between sets), I define a base of an object $A$ to be a set of elements $M$ of $A$ such that for any morphisms $F,G:A\to B$ that ...

**2**

votes

**2**answers

77 views

### Finitary endofunctors: “Support” of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

Consider the word a.k.a. list monad $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$:
...

**5**

votes

**1**answer

156 views

### Reconstruction of coalgebras

In the paper Reconstruction of hidden symmetries of Bodo Pareigis in the subsection "3.1 Reconstruction of coalgebras" there is the following proposition (3.3.).
Let $\mathcal{C}$ be a ...

**3**

votes

**0**answers

107 views

### Reference for duality inducing bijections between subobjects and quotients?

I'm not the most category-theoretic person but I have run into the following statement in my work. Suppose we have a category duality $\mathcal F:\mathcal C\to\mathcal D$. For my needs you can ...

**3**

votes

**0**answers

121 views

### External tensor product Calabi-Yau DG categories

Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...

**2**

votes

**0**answers

58 views

### Closed embedding into a normal Hausdorff space and left lifting property

I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a ...

**5**

votes

**1**answer

171 views

### Can we construct a filtered chain complex from a spectral sequence?

Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...

**2**

votes

**0**answers

136 views

### Small abelian categories and module categories - preservation of injective and projective objects

A soft question on small abelian categories:
https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper
Wikipedia: "The article "Sur quelques points d'algèbre homologique" by ...

**12**

votes

**2**answers

476 views

### Reference request for Linton's theorems on equational theories

This is a reference request for the following "well-known" theorems in category theory:
There is an equivalence of categories between finitary monads on $\mathbf{Set}$ and finitary Lawvere ...

**18**

votes

**0**answers

375 views

### Does the Ackermann function count something?

Let $\mathrm{FinSet}$ be the category of finite sets.
A finite set structure is a faithful functor $F\colon C\to \mathrm{FinSet}$ such that, for any $n\geq 1$, there are only finitely many isomorphism ...

**5**

votes

**1**answer

243 views

### Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself?

Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, ...

**3**

votes

**0**answers

95 views

### Inductive limit of inclusions

Let $(\Lambda, \le)$ be a directed system and $\{ X_{\alpha} \}_{\alpha \in \Lambda}$ be a family of topological spaces indexed by $\Lambda$ such that $X_{\alpha} \subseteq X_{\beta}$ whenever $\alpha ...

**2**

votes

**1**answer

159 views

### Does the set of composable arrows in a category have to be a pullback?

When defining a category, do we need to have the set of composable arrows be a pullback of the domain and codomain selecting functions? What can go wrong if we use a subobject of the pullback?
This ...

**3**

votes

**1**answer

125 views

### Sheaves on sites given by a (regular) cd-structure

Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see https://ncatlab.org/nlab/show/cd-structure or Voevodsky's paper Homotopy theory of simplicial presheaves in ...

**5**

votes

**1**answer

214 views

### Yoneda map for a composition of a representable functor and an arbitrary functor

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So ...

**6**

votes

**0**answers

106 views

### 2-fibrations in terms of comma categories

Fibrations of 2-categories and bicategories were defined by Hermida and Bakovic, respectively, and their Grothendieck construction was studied by Buckley in Fibred 2-categories and bicategories. ...

**1**

vote

**1**answer

230 views

### Structure/object versus material/aggregate in category theory?

Is there an analogue in category theory of the distinction between structures/objects and material/aggregate, a distinction reflected in the grammar associated with count and non-count nouns in ...

**7**

votes

**1**answer

213 views

### Is $\operatorname{Fun}^\text{small}(\operatorname{Fun}(C,\mathsf{Set}),\mathsf{Set})$ total when $C$ is small?

Let $\mathcal C$ be a small category. Then it is known that the category $\operatorname{Fun}^\text{small}(\operatorname{Fun}(\mathcal C,\mathsf{Set}),\mathsf{Set})$ of functors $\operatorname{Fun}(\...

**30**

votes

**3**answers

1k views

### Shapes for category theory

Most texts on category theory define a (small) diagram in a category $\mathcal{A}$ as a functor $D : \mathcal{I} \to \mathcal{A}$ on a (small) category $\mathcal{I}$, called the shape of the diagram. ...

**14**

votes

**2**answers

521 views

### Recovering an abelian category from the Ext of its simple objects

Let $C$ be an abelian category, assume for simplicity that $C$ is enriched over $Vect_k$ (vector spaces over $k$) for some fixed field $k$.
Suppose also that $C$ is both Artinian and Noetherian, so ...

**7**

votes

**1**answer

176 views

### Categorical semantics of universe levels in dependent type theory

I know that locally closed cartesian categories provide categorical semantics for type theories with dependent products.
What kind of categories model type theories with infinite universe hierarchies (...

**2**

votes

**0**answers

149 views

### Categorical setting for cancellation in direct sums

I am wondering whether some criterion can be put on a category $\mathcal{C}$ with direct sums to ensure that for three objects $X,Y,Z$ one has
$$
X\oplus Y \cong X \oplus Z\Longrightarrow Y\cong Z.
$$
...

**8**

votes

**3**answers

351 views

### Which categories are injective with respect to fully faithful functors?

Recall that a poset $K$ is a complete lattice if and only if $K$ is injective with respect to poset embeddings in that sense that for any poset $B$, any embedded subposet $A \subseteq B$, and any ...

**14**

votes

**2**answers

1k views

### How many category structures are possible on two sets?

For two sets $O$ and $A$, we will call a category structure a collection of functions ${\sf dom}:A\to O,\ {\sf cod}:A\to O,\ {\sf 1}:O\to A,\ \circ:A\times_OA\to A$ satisfying the usual axioms for a ...

**3**

votes

**0**answers

234 views

### Cyclic lists of multisets

I am wondering if it is appropriate to ask about having a specific algebra for an endofunctor computed. We all know about the multiset monad, and it's endofunctor $\mathcal{M}_S$, and some might know ...

**9**

votes

**0**answers

153 views

### Aspherical fibrations and group epimorphisms

Let $\mathsf{Top}$ denote the category of pointed spaces having the pointed homotopy type of a pointed CW-complex. Let $\mathsf{Grp}$ denote the category of groups. It is well documented that for ...

**4**

votes

**0**answers

63 views

### Braided monoidal categories as doubly degenerate tricategories

The fact that any tricategory with a single 0-cell and a single 1-cell is a braided monoidal category seems to be widely known. For instance, it appears in the "periodic table" of n-...

**7**

votes

**0**answers

160 views

### How much is known about the consistency strength of toposes and topos-like categories?

It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ...

**1**

vote

**0**answers

95 views

### The difference between Agda and Idris for programming using Homotopy type theory [closed]

Which is better for programming benifiting by Homotopy type theory(HoTT),Idris or Agda.compare them.

**8**

votes

**1**answer

171 views

### Filtered 2-colimits commute with finite 2-limits

Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only ...

**9**

votes

**1**answer

395 views

### Is there a large colimit-sketch for topological spaces?

Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$?
In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}...

**11**

votes

**1**answer

328 views

### Yoneda Lemma for monoidal functors

Let $(\mathcal V,\otimes,I)$ be a closed symmetric monoidal category, and let $\mathcal C$ be a $\mathcal V$-enriched category. The (weak) enriched Yoneda Lemma gives us a nice description of the set $...

**6**

votes

**0**answers

224 views

### Which properties of categories are preserved under homotopy equivalences?

This question arose from a discussion in the comment section of another MathOverflow question with Mike Shulman and Alec Rhea, which raised the following point:
Vague Question: Are adjunctions an ...

**2**

votes

**0**answers

37 views

### Relationship between connectedness and (co)separators

What is known about the relationship between connectedness and (co)separators?
A category is said to be connected iff it is inhabited and every object is connected by a zigzag of morphisms to every ...

**35**

votes

**12**answers

3k views

### No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...

**5**

votes

**1**answer

137 views

### Enriched coends which preserve equivalences

Although this question might be formulated in higher generality, let me try to be concrete:
Let $(\mathbf{Top},\times,*)$ be the monoidal category of compactly generated weak Hausdorff spaces; and let ...

**6**

votes

**0**answers

174 views

### Different levels of isomorphism/equivalence/adjunction between bicategories

What are all the different levels of 'isomorphism/equivalence/adjunction' we can have between bicategories? Do any of them 'collapse' to one-another?
When working with $1$-categories, we have four ...

**2**

votes

**0**answers

59 views

### Left anodyne is covariant equivalence

I have a question about the proof of left anodyne between two simplicial sets over $S$, where $S$ is a simplicial set, is covariant equivalence. In the proof (HTT 2.1.4.6 or https://ncatlab.org/nlab/...

**6**

votes

**1**answer

242 views

### $\infty$-natural transformations and adjunctions

I'm having troubles proving these two related statements, which are immediate for 1-categories and should of course be true for $\infty$-categories:
Given a natural transformation $\alpha: f \...

**0**

votes

**0**answers

88 views

### Factoring a natural transformation through a functor

This seems fairly similar to density. Suppose I have three categories $A,B,C$, and a functor $L: B \to C$ so that every natural transformation $f: L.F \Rightarrow L.G$, for a parallel pair $F,G: A \to ...

**2**

votes

**1**answer

154 views

### On the functors $\text{Hom}_R(k,-)$ and $k \otimes_R ( -)$ for Artinian local Gorenstein ring $R$

Let $(R, \mathfrak m,k)$ be an Artinian local Gorenstein ring, hence $\text{Hom}_R(k, R)\cong k$, and so
$\text{Hom}_R(k, R^{\oplus n})\cong k^{\oplus n} \cong k \otimes_R R^{\oplus n} , \forall n \ge ...

**8**

votes

**0**answers

399 views

### Ring of global sections of a functor $\mathbf{CRing} \to \mathbf{Set}$

Let $U : \mathbf{CRing} \to \mathbf{Set}$ be the forgetful functor. For any functor $F : \mathbf{CRing} \to \mathbf{Set}$ consider the class of natural transformations
$$\mathcal{O}(F) := \mathrm{Hom}(...

**6**

votes

**0**answers

234 views

### Examples of (co)ends

I am reading texts about (co)ends, and everywhere I see a lack of examples. I am not an expert in this area, and without examples it is difficult for me to use my intuition to grasp the idea. MacLane ...

**2**

votes

**0**answers

76 views

### Freeness of the action of the ground monoid in a monoidal category

Let $(\mathcal{C}, \otimes , 1)$ be a monoidal category, and let $\mathrm{End}_{\mathcal{C}} (1)$ be the ground monoid of $\mathcal{C}$ - which is a commutative monoid. If $r_X : X \otimes 1 \to X$ ...

**4**

votes

**0**answers

129 views

### What is the category of homoiconicity?

Category theory often captures the hearts of computer scientists because of its uncanny capacity to cleanly describe CS concepts. A common example is the way Algebraic Data Types can be represented as ...

**9**

votes

**2**answers

287 views

### Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$

Connes showed in Cohomologie cyclique et foncteurs $Ext^n$ (1983) that the classifying space of his cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty = B(S^1) = K(\mathbb Z,2)$.
Connes' proof is ...

**3**

votes

**0**answers

90 views

### Automorphism of a stack morphism

Let $X$ be an algebraic stack and let $f: S \to X$ be a smooth covering of $X$ by a scheme $S$.
Motivation: Forgetting about stacks for a moment and going back to covering spaces: Given a covering map ...

**4**

votes

**1**answer

115 views

### Universal property of the codomain fibration

Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it ...

**3**

votes

**0**answers

26 views

### Lax algebras for pseudomonads and monads in Kleisli bicategories for the induced pseudocomonad

In Day–Street's Lax monoids, pseudo-operads, and convolution, they remark without proof:
There are general principles involved here. Suppose $(T, m, j)$ is a pseudomonad on any bicategory $\mathcal K$...