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.

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11
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0answers
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 $$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$ ...
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1answer
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 ...
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2answers
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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
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1answer
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 ...
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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 ...
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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 ...
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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
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1answer
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, ...
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0answers
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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
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2answers
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 ...
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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
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1answer
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, ...
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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
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1answer
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
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1answer
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
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1answer
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 ...
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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. ...
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1answer
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
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1answer
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}(\...
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3answers
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
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2answers
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
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1answer
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
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0answers
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
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3answers
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 ...
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2answers
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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
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0answers
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
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0answers
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
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0answers
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
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0answers
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 ...
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0answers
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
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1answer
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
1answer
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
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1answer
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 $...
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0answers
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 ...
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0answers
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
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12answers
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
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1answer
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
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0answers
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
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0answers
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
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1answer
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 \...
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0answers
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
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1answer
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 ...
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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2answers
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
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0answers
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
1answer
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
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0answers
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 $\mathcal K$...

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