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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Internal additions in additive categories agree with given ones

I know that if $\mathcal C$ is a semiadditive category, then for every two objects $A$, $B$, the set $\mathrm{Hom}_\mathcal{C} (A, B)$ is automatically endowed with a structure of commutative monoid, ...
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Cauchy completion in ordinary category theory

In Borceux and Dejean's paper Cauchy completion in category theory, they conclude the smallness of $\bar{C}$, the full subcategory of $[C^{op},\mathrm{Set}]$ spanned by all the retracts of the ...
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33 views

Duality in abelian categories.

Let $C$ be an abelian category, with a projective separator $k$. Assume that $C$ has a duality, that's a functor $\ast:C\to C^{\text{op}}$ together with a natural isomorphism $\tau:1\to \ast\ast$ such ...
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51 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
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192 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
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23 views

When can we complete bundle morphisms when given just a morphism of total spaces?

Consider a bundle $E \rightarrow B$ in some category. The morphisms of it to some other bundle $E' \rightarrow B'$ is simply two morphism $E \rightarrow E'$ and $B \rightarrow B'$ which make the ...
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29 views

EXAMPLE on p.58, in Hungerford(GTM)

In line 6, there is an explanation, '$h$ is an equivalence in $E$ if and only if $h$ is an equivalence in $C$. ' Suppose that $h : B \rightarrow D$ is an equivalence in $C$. Then $h$ is an ...
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80 views

Criteria for a contravariant functor from Schemes to Sets to be representable by a scheme

I am trying to prove that a functor $\mathcal{F}:(Sch/S)^{op}\rightarrow (Sets)$ is representable by an $S$-scheme $F$. My intuition says that it is indeed representable. I have been reading Fulton's ...
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93 views

Every monoidal category is strictly equivalent to a strict monoidal category.

I am reading Joyal and Street's article "braided tensor categories". The following theorem is proved (Theorem 1.2). Let $ C $ be a category. Let $FC$ and $F_sC$ be respectively the free monoidal ...
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70 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
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60 views

A question about the category of co-elements

The category of elements of a $K: \mathcal{D} \rightarrow \mbox{Set}$, denoted by $1\downarrow K$ is defined to be a category which has as objects arrows of the form $d: * \rightarrow Kd$ i.e. the ...
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97 views

Definition for a bar resolution for a module over a dg category

Let $ \mathcal{A}$ be a dg category and define a right $ \mathcal{A}$ module to be a dg functor $ M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
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39 views

A full embedding in a finitely complete category

I know that each category can be full embedded in a complete quasicategory. My question is: a category can be full embedded in a finitely complete category? There is a universal such one (that's a ...
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30 views

Do colimits preserve free group actions?

Suppose $C$ is a small category, $M$ is a category and $G$ is a discrete group. Let $X$ be a $G$-object in $M^C$ and suppose that the action of $G$ on $X$ is object-wise free. Under what conditions is ...
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30 views

Endofunctor as a presheaf

This looks pretty obvious, but an endofunctor $F : Set \to Set$ seems to be a presheaf over $Set^{op}$. Is there any useful fact that can be learnt from this view of endofunctors?
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43 views

Why are there no naturality condition in definition of exponential in a category?

In chapter 6 "Exponentials" of "Category Theory" Second Edition, Steve Awodey, 2010, there is no naturality requirement in the definition of exponential in a category. Exponential is an adjoint and ...
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71 views

Is the category of topological spaces coregular?

Everything is in the title. The category Top of topological spaces and continuous mappings is not regular, but is it coregular ? Furthermore, Top isn't cartesian closed, but does it satisfy the dual ...
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157 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
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46 views

Is the requirement that every morphism factors as an epi composed with a mono part of the definition of an abelian category?

Hilton and Stammbach require an abelian category to be an additive category in which 1) all kernels and cokernels exist 2) all monos are the kernel of their cokernel, all epis are the cokernel of ...
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147 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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37 views

Do graded modules form an additive category?

I got two conflicting references. On page 466 of Shastri's Algebraic Topology, he claims that graded modules with homogeneous morphisms form an additive cat, whereas Hilton and Stammbach say on page ...
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62 views

About details of the Fakir theorem proof (associated idempotent triple)

On the ncatlab work http://ncatlab.org/toddtrimble/published/Associated+idempotent+monad+of+a+monad Todd Trimbe quote the Fakir theorem about the associated idempotent triple, and this is based on ...
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78 views

Is there such a thing as “combinatorial category theory”?

According to wikipedia, In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations Is there such a thing as ...
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98 views

Does $\mathsf{Man}$ possess countable products?

In our lecture we defined a category $\mathcal C$ to have arbitrary products, iff every diagram $A:\mathcal J \to \mathcal C$ with $\text{Morph}(\mathcal J)=\{\text{id}_j\}_{j \in Ob(\mathcal J)}$ has ...
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120 views

How to prove the equivalence of the two definitions of “equivalence of categories”?

As far as I know, there are two definitions of "equivalence of categories". The first definition of the equivalence of categories $A$ and $B$ is required that there exist functors $F\colon A \to B$ ...
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93 views

Is there a topos in which the natural numbers object are the finite dimensional vector spaces?

I recall reading somewhere that there is a topos in which the Dedekind reals are exactly the measurable functions. Now vector spaces are prominently characterised by dimensionality. This prompted the ...
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54 views

Stable fiber products of commutative rings

Let $R_1 \to T$ and $R_2 \to T$ be homomorphisms of commutative rings. Consider the fiber product $R=R_1 \times_T R_2$. Let $R \to R'$ be a homomorphism of commutative rings, and define $R'_i$ to be ...
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75 views

Tensors and Equalizers (or Symmetric tensors)

Quick question: Let $(\text{Sym}(V^{\otimes n}), e)$ be the equalizer of all the $S_n$ braidings (that is for $\sigma \in S_n$ we have $a_1\otimes\cdots\otimes a_n \mapsto a_{\sigma 1} \otimes ...
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54 views

Is $\Delta$ dense in $\bf Cat$?

I'm trying to prove that the functor $i\colon\Delta\to{\bf Cat}$ which regards $[n]\in\Delta$ as a category is dense, to deduce that the nerve $N\colon\bf Cat\to sSet$ is fully faithful: how can I do? ...
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26 views

Are lax kernel pairs stable under change of base?

The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps ...
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70 views

Associated sheaf functor for sheaves over complete Heyting algebras

Suppose $A$ is a complete Heyting algebra and $a : \widehat{A} \to \text{Sh}(A)$ is the associated sheaf functor (where the topology on A is the usual sup topology). Is there a simpler description of ...
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53 views

When is Kleisli category regular?

Are there any known conditions on the monad $\mathbb{T}$, such that $\mathcal{C}_{\mathbb{T}}$ (in particular when $\mathcal{C} = \mathbf{Set}$) is regular?
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101 views

What is the coimage of a ring homomorphism?

Is there a general way to find the coimage of a ring homomorphism? For example, for the canonical injection $\mathbb{Z}\rightarrow\mathbb{Q}$, the image is $\mathbb{Z}$ but the coimage is ...
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How to denote the set of binary relations of which a particular ordered pair is a member?

Given a universe $U$ and two subsets $S$ and $T$ (also, both members of $U$), what is the name given to denote the set of all binary relations in $U$ where the ordered pair $(S,T)$ is a member? The ...
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87 views

When terminal objects are separators?

In category of Set the terminal object is also a separator (or generator). But this is not correct in general category. What condition would guarantee that the terminal object of a category if exists ...
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43 views

How to pose a decent counting problem of cells in higher category?

Suppose you are given a power series $$S = \Sigma_{i=0}^{\infty}a_ix^{i}$$ with coefficients in $\mathbb{N}$, and you are tasked with telling if there can not be a finite category (or any kind of ...
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Categories of sheaves of finite homological dimension revisited

This is the question When is the category of (quasi-coherent) sheaves of finite homological dimension? revisited. I made some mistakes: I posted the question before I sign in, I tried to make it more ...
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119 views

Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd). Questions: Suppose that we ...
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252 views

Concrete category and “abstract category”

From Analysis and Its Foundations By Eric Schechter A concrete category consists of a collection of objects and a collection of morphisms. I am curious what an "abstract category" is? I found ...
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78 views

Picturing resolutions of complexes

I got a question about resolutions of complexes, I just wanted to make sure I'm looking at them the right way. Let $\cdots \rightarrow P^{-1} \rightarrow P^{0} \rightarrow X \rightarrow 0 ...
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66 views

Inductive vs projective limit of sequence of split surjections II

This question is a follow-up of this earlier question I asked. Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of ...
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40 views

Formulating Cauchy Completeness in the category of linear spaces using categorical limits and colimits

Formulating Cauchy Completeness in the category of linear spaces using categorical limits and colimits, can it be done? I've made some naive attempts replace cauchy sequence with sum and taking the ...
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42 views

Name for the initial object of a connected component of a category

I was just wondering whether there is already a name for an object of a category as described below. Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that ...
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Quivers with a binary operation on the arrows

A set with an arbitrary binary operation is called a magma. A set of dots with a set of arrows between them is called a quiver. A category is a quiver with a binary operation on the arrows obeying ...
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Involutive Functors

Let us call a (co)functor $F:\mathcal{C}\to\mathcal{C}$ on a category $\mathcal{C}$ involutive, if $F^2$ is naturally isomorphic to $1_\mathcal{C}$. For example, if ...
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Fibered Product of Subcategories

Is there a general construction or existence theorem for the fibered product of two subcategories of some ambient category? What sort of problems might one run into? Does this require a 2-categorical ...
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83 views

Thompson's group F and monoidal categories

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category generated by one object $A$ such that there exists an isomorphism $\alpha: A \otimes A \to A$, then for every object $X ...
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Morphisms with an arbitrary number of objects

Is this structure familiar for you? It consists of a category $C$ a set $M$ a function ``$\operatorname{arity}$'' defined on $M$ a function $\operatorname{Obj}_m$ defined for every ...
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166 views

Applications of monads in general topology?

What are applications of monads in general topology? For example, for GT is important the notion of products, products are adjoints, so adjoints may be important for GT, but what's about monads?
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The Two Eilenberg-Moores

So, there is the Eilenberg-Moore spectral sequence, and there is (for any monad $(T,\mu,\eta)$ on a category $C$) the Eilenberg-Moore Category $C^T$ of $T$-algebras. The silly question, is the ...