# Questions tagged [ct.category-theory]

Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

5,897
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### Is there a universal property characterizing the category of compact Hausdorff spaces?

This is in some sense a follow up to the question asked here Properties of the category of compact Hausdorff spaces
To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\...

0
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0
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150
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### On the definition of small categories in SGA4

We assume ZFC+U.
A category is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying ...

6
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2
answers

451
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### Overloading of the word "local" in category theory

The word "local" in category theory does not seem to have a precise definition in itself but it often appears as part of other terminology. To my understanding, it is then used in the ...

6
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0
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### $(\infty,1)$-topoi generated by $(n,1)$-categories

A (1,1)-topos (i.e. an ordinary Grothendieck topos) is called localic if the following two equivalent conditions hold:
It is the category of sheaves on a (0,1)-site with finite limits$^*$ (i.e. a ...

5
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0
answers

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### Which properties of the pullback/restriction functor imply surjectivity of a ring homomorphism

Let $f:R\to S$ be a homomorhpism of (noncommutative) algebras over some field k (say the complex numbers) and let $F:Rep(S)\to Rep(R)$ be the corresponding functor between the categories of finite-...

4
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0
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### Equivalence of Yetter-Drinfeld modules to Drinfeld center: is there a purely categorical proof?

Let $H$ be an Hopf algebra over a field $k$, and let $\mathcal{C}$ be the monoidal category of left $H$-modules.
It is known that the Drinfeld center of $\mathcal{C}$ is equivalent (as a braided ...

1
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1
answer

136
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### Categories associated to digraphs

Let's take a directed graph, or a digraph, $G=(V,E)$ given by a finite set of vertices $V$ and a finite set of edges $E$. We can assume that pairs of vertices can have parallel edges between them and ...

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### $(\infty,2)$-categories as colimits of orientals

Let $\mathcal{C}$ be an $\infty$-category represented by a fibrant simplicial set in the Joyal model structure. It is well known that $\mathcal{C}$ can be expressed as the (homotopy) colimit over its ...

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0
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### Why is the induced singleton pretopology closed under pullbacks?

Let $(C,\mathcal{T})$ be an arbitrary site with a pretopology $\mathcal{T}$. The category $C$ has coproducts.
As a pretopology I mean the definition 2.24 of Grothendieck topology
in Angelo Vistoli’s ...

2
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0
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### Category with two types of morphisms and a weak transitive like relationship between them

Take a category A that has two types of morphism structures on it (can think of it as two separate categories over the same objects) which we call type $a$ and type $b$. Then if $g_b f_a$ exists there ...

2
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2
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### Can a category be enriched over abelian groups in more than one way?

An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way?
An abelian ...

10
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0
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### Rectifying diagrams of $\infty$-categories

If $C$ is a 1-category and $D$ is a locally presentable $(\infty,1)$-category presented by a combinatorial model (1-)category $M$, then any $(\infty,1)$-functor $C\to D$ can be represented by a strict ...

5
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1
answer

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### Fibrations of sites for $\infty$-topoi

For any geometric morphism $f:\mathcal{F} \to \mathcal{E}$ of Grothendieck 1-topoi, there exists a functor of small categories $\ell :D\to C$ and left exact localizations $\mathcal{F} \hookrightarrow \...

7
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1
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### Which direction does a lax dinatural transformation go?

In 2-category theory we often encounter notions with noninvertible coherence 2-cells, in which case there is a choice about which way the 2-cell should point. Generally, one of these directions is ...

17
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1
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### Small complete categories in HoTT+LEM

Freyd's theorem in classical category theory says that any small category $\mathcal{C}$ admitting products indexed by the set $\mathcal{C}_1$ of all its arrows is a preorder. I'm interested in whether ...

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0
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### Can the Haskell kind system be interpreted as a category?

Background
I consider myself fairly knowledgeable about the parts of category theory that concerns computer science, and while I have seen many examples of how Haskell kind constructors can be thought ...

2
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0
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### When a monoidal closed category is cartesian closed

Let $C$ be a monoidal closed category with tensor $\otimes$ and internal hom $[-, -]$.
Suppose that
$C$ acts by adjoint monads, i.e. $- \otimes X$ is a comonad and $[X, -]$ is a monad, and each $F : ...

10
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1
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495
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### Universal property of the set of injections in the category of sets

Given two sets $A$ and $B$, the function set $B^A$ is characterized by the universal property that the functor $(-)^A:\mathrm{Set} \to \mathrm{Set}$ is the right adjoint of the functor $(-)\times A:\...

6
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0
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### Is the category of projections interesting?

Let $C$ be a category and let $C'$ be the wide subcategory whose maps are projections, that is maps in $C$ which belong to some limiting cone (over a discrete base). Since limiting cones compose, $C'$ ...

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0
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### Name for homotopy totalization of Goodwillie tower (in embedding calculus)

Let $M,N$ be a manifold and consider the presheaf of spaces $\textrm{Emb}(-, N)$ on the open sets of $M$. Classical embedding calculus produces a goodwillie tower
$$ \ldots \rightarrow T_{k+1} \textrm{...

5
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1
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194
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### Simple type theory: equational axioms validated by biCartesian closed categories

In this question, we consider only type theories with no ground types and no function symbols.
I want to know whether there exists a model of simple type theory with finite products, finite coproducts,...

5
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1
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300
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### On the definition of infinity-category

On 8:38 of Session 9: Masterclass in Condensed Mathematics an $\infty$-category is defined as a simplicial set $\mathcal{S}$ (i.e a functor $\Delta^{op}\rightarrow Sets$) such that for every horn $\...

2
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0
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### Cat as a bicategory of monads over another category

Let's assume infinitely many Grothendieck universes exist. Let's call $\kappa$-Cat the bicategory of $\kappa$-small categories with anafunctors and anatural transformations. Now for any $\lambda$ and ...

5
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1
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### Lift a monad along a generic right adjoint

$\require{AMScd}$We have a neat way to lift a monad along a monadic right adjoint, through a distributive law: in a setting like
$$
\begin{CD}
X @. X \\
@VUVV @VVUV\\
C @>>T> C
\end{CD}$$
if ...

3
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0
answers

139
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### Kleisli is initial, a coherence issue

Usually, when proving that the Kleisli adjunction of a monad $M$ is initial in the category of all adjunctions splitting the monad, we hide coherences under the carpet.
Or so did I, until I got ...

4
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3
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452
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### "Quasi-coherent" vector spaces in Sch/S

$\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A_S$-vector space objects ...

5
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1
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121
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### Category of multisorted Lawvere's theories

Multisorted Lawvere's theory consists of
sets of sorts $S$
small category $T$
a preserving-product essentially bijective functor $(\mathrm{finSet}/S)^{\mathrm{op}} \to T$
How is category of ...

5
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0
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88
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### Enriched categories from Lawvere theories

Given a Lawvere theory $\mathbb{T}$, is there a convenient way (e.g. a functorial construction) that freely enriches a category $\mathcal{C}$ to a $\mathbb{T}$-enriched category? For example, given ...

4
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0
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### $(\infty,n)$-operads?

I wonder whether there is (or should be) a theory of colored $(\infty,n)$-operads or multicategories?
We know that multicategories are generalizations of categories, and nonsymmetric colored $\infty$-...

6
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1
answer

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### Decategorifying Grothendieck topoi and categorifying topological spaces

(This is in a sense a follow-up to this question.)
I was under the impression these days that Grothendieck topoi were also¹ analogous to topological spaces in that the former were left exact ...

3
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1
answer

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### Is a monad functor also known as a monad map?

Suppose I have a monad $M_S = \langle S , \eta_S, \mu_S \rangle$. I want to map this monad to another monad, $M_Q = \langle Q , \eta_Q, \mu_Q \rangle$. What is the minimum I have to define to give ...

2
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1
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155
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### Is every filtration on an abelian category strict?

It's well-known that the category of filtered objects in an abelian category is generally not an abelian category. One must generally add extra structure to ensure that all morphisms are strict for ...

5
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1
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140
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### Is there a canonical product on the category of monads on Set?

I would like to know if there is a partial monoidal product on the category of monads on Set. I want this partial monoidal product to "handle" monad composition which we understand exists ...

1
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1
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### It's there a way to take a composite monad and a monad map to create a map of the composite?

Let us suppose you start with two monads, $M_S = \langle S , \eta_S, \mu_S \rangle$ and $M_T = \langle T , \eta_T, \mu_T \rangle$ and suppose you have a distributive law, $\lambda: ST \rightarrow TS$ ...

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0
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### Lagrangian subcategories of (non-pointed) braided tensor categories

I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...

15
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2
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### Why do we care about small sets?

I have been a user of category theory for a long time. I recently started studying a rigorous treatment of categories within ZFC+U. Then I become suspecting the effect of the smallness of sets.
We ...

9
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1
answer

373
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### Proof that the unit of a Cartesian monoidal category is terminal

In short, given a monoidal category whose product is the categorical product, show that the unit object is terminal.
This looks very similar to questions that have been answered, but is subtly ...

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1
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### Is there a good theory of 2-locales?

Topological spaces and locales are two closely related notions meant to capture the concept of an abstract space, the latter of which admits a certain "noncommutative" generalisation known ...

8
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2
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### In which categories is the union of subobjects given by the pushout over the intersection?

Let $\mathcal C$ be a sufficiently-complete-and-cocomplete category. Let $C \in \mathcal C$, and let $A \rightarrowtail C \leftarrowtail B$ be subobjects. Let $A \cap B = A \times_C B$ be the ...

3
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0
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### The site and the space

There is a (seemingly simple) statement in the literature on sheaf theory, namely,
If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of ...

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### Category whose morphisms are commutative monoids but not enriched

In a recent investigation, I constructed a category $\mathcal{C}$ with the following property. For objects $X,Y \in \mathcal{C}$, the morphism set $\text{Mor}(X,Y)$ is a commutative monoid with ...

8
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1
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### Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors?

Let $A$ be an abelian category (you can assume additional conditions for its goodness). Let $\mathrm{Seq}(A) = \mathrm{Func}(\mathbb{Z}, A)$, where $\mathbb{Z}$ is the standard order category on ...

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1
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### Is it possible to set up multiple automorphisms over a structureless object inside single-sort defined category?

I was trying to understand the behaviour of the primitive equality (=) in the axiomatization of category, which takes morphisms as primitives and objects as derivatives in bijection to identity ...

8
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### Isbell duality between algebras and sheaves

nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...

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0
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### Morphism factoring through a subobject

Van den Berg in his work on predicative topoi says the following:
"Let C be a category. A map f : B → A is a cover, if the only subobject of A through which it factors is A itself."
I have ...

0
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0
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### How to canonically induce a morphism between module categories of path algebras when given a morphism of quivers?

Let $\pi:P\to Q$ be a morphism of quivers, i.e., $s(\pi(a))=\pi(s(a)),t(\pi(a))=\pi(t(a)) $ for arrows $a$, start $s$ and tail $t$. Is there a canonical way to induce from $\pi$ a morphism between ...

9
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2
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### Is there a generalization of braided monoidal category without the isomorphism requirement?

Is there a generalization of the notion of braided monoidal category that does not force the braiding $\gamma$ to be an isomorphism? I mean, it is of course possible to define such a kind of category, ...

4
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0
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332
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### Are Frobenius modules related to Frobenius algebras?

Frobenius modules appear in the Riemann Hilbert correspondence.
Frobenius algebras appear in TQFT.
Is there a way to pass from one to the other?

9
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0
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### Cocompleteness of enriched categories of algebras

A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch ...

8
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1
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### How many categories $C$ are there such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$?

In this question, bimonadic category is a category $C$ such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$.
How many bimonadic categories are there? Can we classify them all?
...