Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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Are strong monomorphisms coretrations in the category of graphs?

Consider the category of digraphs with strong homomorphisms as the morphisms. Here a strong homomorphism $f: G\longrightarrow H$, is a graph homomorphism which preserves and reflects adjacency of ...
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kernel and cokernel of a morphism in $Map_K$

Ler $Map_K$ the category whose objects are triples $(V,W,f)$, where $V$ and $W$ are finite dimensional $K$-vector spaces and $f:V\rightarrow W$ is a $K$-linear map. A morphism from $(V,W,f)$ to $(V',W'...
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1answer
14 views

Finitely presentable objects and the Kleisli category

There is a correspendence between Lawvere theories $L$ and finitary monads $\mathbb{T}_L$ (associated to $L$), due to Lawvere: the category $Mod(L)$ of models of $L$ (in $\mathbf{Set}$) is equivalent ...
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1answer
16 views

Adjoint to Forgetful Functor from Monoidal Categories to Categories

Let $MonCat$ denote the 2-category of monoidal categories and strict monoidal functors. Let $Cat$ denote the category of 2-categories. There is a forgetful functor $Forget:MonCat\rightarrow Cat$. Does ...
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1answer
29 views

Does the usual filtration on graded objects satisfy a universal property?

Let $\mathsf{A}$ be an abelian category, such as $R \mathsf{Mod}$ for concreteness. We can think of the category of $\mathbb{Z}$-graded objects in $\mathsf{A}$ as the functor category $\mathsf{A}^\...
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Reference request: Categories enriched over $\textbf{Lat}$

I'm looking for some sources that discuss categories enriched over the category $\textbf{Lat}$ of lattices. Actually, more specifically, the category I'm studying is enriched over the category of (...
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40 views

In what sense is metric space completion universal?

The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ...
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1answer
12 views

Cokernel of a module homomorphism

Let $A$ a $K$-algebra. Let $M$, $N$ $A$-modules and $f:M\rightarrow N$ a module homomorphism. The cokernel of $f$ is $Cokerf=N/Imf$ I define a homomorphism $\rho:N\rightarrow N/Imf$ by $\rho(n)=n+Imf$....
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36 views

Fundamental Theorem of Bifunctors

I'm having trouble understanding the proof of the theorem on bifunctors in MacLane's CWM on page 37. Let $\mathcal{C}$, $\mathcal{D}$ and $\mathcal{E}$ be categories and let $ L_y : \mathcal{C} \...
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33 views

Definition of Equivalence of Categories

I was reading about equivalence of categories. So I came across two definitions and they both do not seem equivalent. The first definition is the one given on Wikipedia: Given two categories $C$ and ...
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2answers
38 views

difference between weak and strong actions on a category

If a group acts on a category (in some sense), sometimes the phrases "weak action" and "strong action" come up. I don't know what these mean though. Could someone provide an appropriate definition? (...
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38 views

Slick proof $\mathcal M^\perp$ is limit-stable (strong factorization systems)

Theorem 1.17 of Emily Riehl's Factorization Systems says that given a class of maps $\mathcal M$, $\mathcal M^\perp$ is closed under limits and dually $^\perp\mathcal M$ is closed under colimits. ...
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22 views

Opposite of an undercategory

For a category $\mathcal{C}$ and $*$ a terminal object of it, let $\mathcal{C}_*$ denote the category under this terminal object. Suppose that the opposite of $\mathcal{C}$ is $\mathcal{C}^{\text{op}}...
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1answer
40 views

Is there a relationship between the pullback in differential geometry and that in category theory?

1. Is there a relationship between the pullback in differential geometry and the pullback in category theory? [2. Is there a relationship between the pushforward/pushout in differential geometry ...
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26 views

Exhibit a chain of adjoints passing through the diagonal

A problem in Leinster's Basic Category Theory: Fix a topological space $X$, and write $\mathscr{O}(X)$ for the poset of open subsets of $X$, ordered by inclusion. Let $$\Delta: \mathsf{Set} \to ...
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27 views

relationships of topological and C* concepts in noncommutative topology

According to wikipedia, noncommutative topology is " a term used for the relationship between topological and C*-algebraic concepts". Can somebody expand on this, give examples/theorems/results and ...
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19 views

Separable algebra in tensor category

I am trying to understand the definition of a separable algebra in a tensor category. In the book $Tensor\ Categories$ by Etingof, Gelaki, Nikshych and Ostrik they provide the definition: "Definition ...
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40 views

equivalence of categories between modules and vector space

Let $Map_k$ the category where the objects are triples $(V,W,f)$, where $V$ and $W$ are finite dimensional $K$-vector spaces and $f:V\rightarrow W$ is a $K$-linear map. A morphism from $(V,W,f)$ to $(...
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2answers
69 views

What does “is natural in $A$” mean in this context?

While reading Bredon's Topology and Geometry, I've come across the following claim: Naturality in $A$ of the sequence defining $\text{Ext}(A,G)$ shows that $\text{Ext}(A,G)$ is a contravariant ...
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1answer
34 views

Let $\mathcal{A}$ be an Abelian category (as defined in Stacks), then all monomorphisms are kernels.

I'm struggling to prove this statement, using the definitions below (I'm assuming the proof for the statement about epimorphisms is analogous). I know that a morphism $f:x\to y$ is monic if and only ...
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24 views

Do the effective descent morphisms w.r.t the codomain fibration hint at the “right topology”?

An intuitive approach to basic descent theory for me started with open covers $ \left\{ U_i \right\}$, replaced them with a singleton covering $\coprod _iU_i\rightarrow X$, and then generalized to ...
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31 views

Morphisms as zigzags, composition as concatenation

I am trying to find any references to the following construction that relates to zigzag categories (https://ncatlab.org/nlab/show/zigzag+category). Let C be a category. Let Z(C) be the category with ...
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1answer
57 views

Grothendieck ring of varieties

In the context of Grothendieck ring of varieties there is often used notion of variety over variety (for example here -2.2.1). I always used only varietes over field. My question is : how is it define ...
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39 views

Day convolution on the category of copresheaves on the opposite of a monoidal closed category

When $\mathcal{C}$ is symmetric monoidal closed, the Day convolution gives a symmetric monoidal closed structure to the category $[\mathcal{C},\mathbf{Set}]$. Suppose that instead, it is $\mathcal{C}...
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3answers
46 views

Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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1answer
30 views

Simple question regarding the monad whose algebras are the category of sets

The category $\mathbf{Set}$ of sets can be viewed as the category of models for a Lawvere theory, and hence it is equivalent to the Eilenberg-Moore category of algebras of an (associated to the ...
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1answer
86 views

For which category (if any) are Lie algebras the algebras of a monad?

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...
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1answer
41 views

Initial object is limit of identity functor: converse

A known theorem in category theory is Suppose $\mathscr{C}$ has an initial object $c$. Then $c$, along with its unique maps, forms the limit of the identity functor $\mathscr{C} \to \mathscr{C}$...
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35 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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1answer
54 views

Are functors (from small categories) functions?

I am looking for either (1) confirmation that the following is true, (2) the mistake making it false pointed out to me: Let $F:\mathcal{C} \to \mathcal{D}$ be a functor from a small category $\...
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102 views

Is there a general way to tell whether two topological spaces are homeomorphic?

We know that if two topological spaces $X$ and $Y$ are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \...
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Diagonal Functor an Isofibration?

Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor. Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any ...
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36 views

Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
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3answers
100 views

“Alternatives” to Natural Transformations

I would like someone to either (1) point out the mistake in what follows or (2) confirm what is said is correct. This would be accomplished by addressing the part in yellow only. The rest of the ...
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Are exponential objects examples of (co)universal objects which are not (co)limits?

This is in some sense a follow-up to previous questions I have had asking about the relationship between products and exponential objects. Products can be written as, and in often are defined to be, ...
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0answers
30 views

Is the tensor product of BAOs a kind of extended BAO?

I've been reading "Boolean algebras with operators. Part I." (Jonsson, Tarski) where, given a subalgebra of a Boolean Algebra, they define its perfect extension. As far as I understand it can be ...
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3answers
45 views

A question about the product functor on finite sets

I am a beginner in Category Theory so please excuse me if this is a trivial question. Let $\mathbf{FSet}$ denote the category of finite sets. The product functor $X\times -:\mathbf{FSet}\to \mathbf{...
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3answers
60 views

Is there a mathematical difference between currying and partial application?

I know the following example doesn't make what I am saying rigorous, but hopefully it clarifies to some extent what I mean. For various computer implementations, dividing by 2 and multiplying by 0....
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1answer
53 views

Evaluating a colimit of functors

This is my first question here. I am working on some categories and this question regarding functors seemed very natural: Let $\{G_\alpha \rightarrow F\}_{\alpha \in \Gamma}$ with $F$ and $\{G_\alpha\...
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2answers
66 views

Categorical Interpretation of Strongest/Weakest Topology

One way to define the product topology is as the weakest topology for which all projection maps are continuous. Strongest/weakest topologies satisfying a given property are ubiquitous in topology and ...
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54 views

Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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1answer
53 views

Algebras with a self-dual congruence lattice

The well known (Mal'tsev) conditions that characterize certain properties of the congruence lattice of an algebra. The existence of a 3-ary term $p$ together with familiar identities $p(x,y,y) \sim x \...
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1answer
61 views

Relationships between categories of elements giving relationships between the functors

Suppose $\mathscr{C} \underset{G}{\overset{F}{\rightrightarrows}} \mathsf{Set}$ are such that $\operatorname{El}F$ and $\operatorname{El}G$ have initial objects which lie in fibers over the same ...
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10 views

How to show these two definitions of Moss' language are equivalent? (Coalgebraic logic)

Definition 1: Suppose $T:\mathbb Set \to \mathbb Set$ is set-enofunctor that preserves weak pullbacks. The $\kappa$-Moss' language, written by $\mathcal L_T^\kappa$ is the carrier set of $(\mathcal ...
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Why care about relation liftings under (covariant) power-set functor? (coalgebraic logic)

Moss' motivation to use the notion of relation lifting was in their importance when applied to membership relation, which leads to define the semantics of $\nabla$ operator. Now in the whole process, ...
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1answer
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How to show the $\kappa$-small functor is $\kappa$-accessible? (coalgebraic logic)

A $\mathtt {Set}$-functor $T:\mathtt {Set} \to \mathtt {Set}$ is defined to be $\kappa$-accessible for a regular cardinal $\kappa$ iff for all sets $X$ and all $x\in TX$ there exists a subset $Y\...
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1answer
35 views

Categorical definition of closure operation

Consider a finitely complete category $\mathscr{B}$. A universal closure operation on $\mathscr{B}$ consists in giving, for every subobject $S\rightarrowtail B$ in $\mathscr{B}$, another subobject $\...
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1answer
41 views

Circular definition in slice category?

I am reading Aluffi (Algebra Chapter 0) there he introduces the slice category in a kind of excercise: When thinking about it I got confused about the "nature" of the $Z$ (and $A$). Since they are ...
4
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1answer
55 views

What is an injection in a topos?

I am reading the paper of Andrej Bauer that proves there is a realizability topos in which there is an injection from the internal Baire Space $\mathbb{N}^\mathbb{N}$ to the natural numbers $\mathbb{N}...
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2answers
45 views

Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...