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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Categorical presentation of “the theory of structure in Set”

I have been thinking about colimits in finitely accessible categories. Here is a paper that abstracts the notions of Domain theory up to categories themselves. This means that we have notions of ...
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Is category theory constructive?

Roughly: This question concerns the process and the constructive nature of formalizing and proving category theoretic statements within $\textsf{ZFC}$. $\textsf{ZFC}$ can only talk about sets, ...
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Consecutive compositions in exact triangles are zero

1) In Weibel's Homological Algebra the definition of a triangle $$A \overset{u}{\to} B \overset{v}{\to} C \overset{w}{\to} T(A)$$ does not include the condition that $vu, wv, T(u)w = 0$ and the ...
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What is the difference between a functor that commutes with limits and a functor that preserves limits?

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between two categories $\mathcal{C}$ and $\mathcal{D}$ where the notion of $F$ preserving and commuting limits makes sense. I am unable to understand the ...
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62 views

Empty set as limit

Let $\mathbf{C}$ be a small category and $C \in \mathbf{C}$. We can construct a limit in $S$ in $\mathbf{Hom}(C, -): \mathbf{C} \to \mathbf{Sets}$ as follows $$ S = \left\lbrace k \in \prod_{D \in ...
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Coherence results

As far a I know, there are two kinds of results which are called 'coherence theorems'. For bicategories they take approximately the following forms: Every diagram of a certain form commutes A ...
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The terminal object comes for free in the definition of a subobject classifier

This is Fact 1.4 in Tom Leinster's informal introduction to topos theory. It states the following: if there exists a mono $t:T \hookrightarrow \Omega$ that classifies the monos in our category, in the ...
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Are these endofunctor categories compactly accessible? (Given a suitable base…)

Here we see a definition of compactly accessible categories. Endofunctor categories have a monoidal product given as functor composition. Let us take an object $O$ in the base category $C$. Now ...
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70 views

Hochschild (co)homology and derived functors

Suppose that $A$ is (complex) unital algebra. We will consider $A-A$ bimodules $M$: such a bimodule is the same as (say) left $A \otimes A^{op}$ module. Let us define $C_n(A,M)$ as $M \otimes ...
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50 views

Why mention the “self-conjugate” property in Tannaka duality?

Based on this Wikipedia section and this MathOverflow answer of Qiaochu, I believe I've understood Tannaka duality for finite groups. We wish to characterize a finite group $G$ as a subgroup of ${\rm ...
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83 views

Different interpretations of a monoid as a category

What is the relation between the categories $\mathbb{N}_0$ and $\mathbb{N}'_0$ as follows: Both objects and arrows of $\mathbb{N}'_0$ are the natural numbers and f is an arrow $f:a\to b$ iff $f+a=b$. ...
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Limits in category of cones.

I'm trying to do exercise 2.17.2 in Borceux's "Handbook of Categorical Algebra": Consider a functor $F: \mathfrak{D} \to \mathfrak{C}$ and the category of cones on $F$. Show that $F$ has a limit if ...
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monomorphism in category theory.

Why if $kernel f=zero morphism$ then $f$ is monomorphism? The converse is very easy,but for this i'm tried to find sufficient condition to equivalent of two morphism(to use the definition of ...
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71 views

Is tilting theory extended also to arbitrary derived categories?

I was reading papers by Rickard ("Morita theory for derived categories") and Keller ("Derived categories and tilting") on tilting theory in derived categories, they seem to focus mostly on module ...
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31 views

An “identity” functor $f:\mathbf{Rel} \to \mathbf{Rel}^{OP}$

Looking at the category $\mathbf{Rel}$ and its opposite, I would like to know if there is something I'd call identity functor, $f:\mathbf{Rel} \to \mathbf{Rel}^{OP}$ that sends a set to itself and ...
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164 views

How to intrinsically think about simplicial objects.

It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps ...
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Associativity of the tensor product of dendroidal sets

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
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140 views

Homology and (co)Limits

I've looked around on MSE and online only to find scattered results, which confuse me. I want to understand how homology behaves with (co)limits. I want to know in particular about singular homology, ...
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32 views

What do we call the construction of a one object $n+1$-category (etc.) from multiobject $n$-category (etc.)?

In general, when we have an $n$-something ($n$-category, etc.) and we decide to push everything up a notch, so $n$-cells become $n+1$-cells, and we end up with a $1$-object $n+1$-something, what do we ...
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Reference for Algebraic Categories

There seems to be a useful result saying the forgetful functor from algebraic categories reflects limits and filtered colimits. My problem is that I haven't been able to find a source which both ...
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89 views

Name for categorical product inside a monoid

If a monoid is a category with a single object, is there a "monoid-theoretical" concept that the categorical product translates to? As an analogue, in a poset the product translates to the notion of ...
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27 views

On the coequaliser of a kernel pair

How does one prove the following statement about kernel pairs? If a pair of parallel morphisms is a kernel pair and has a coequalizer, then it is the coequalizer of its kernel pair.
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51 views

What can be said about the terminal object in a category of pullbacks?

Given a category $A$; consider the category of arrows $A^2$, whose morphisms are commutative squares, which are further pullback squares. Suppose this category has a terminal object $a \rightarrow b$. ...
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Right cancellation of homomorphisms on groups implies

Let $f:G \to H$ be a a group homomorphism such that for any two groups $H_1 , H_2$ , and any homomorphisms $g_1 : H \to H_1 , g_2 :H \to H_2$ , $g_1 \circ f = g_2 \circ f \implies g_1=g_2 $ ; then is ...
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Why is it difficult to define n-category?

Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally. If we have a definition for 1-categories (category) and a definition for ...
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small limits in the pointed category of a category with all small limits

In Hovey, Model Categories, he says that given a category $\mathcal{C}$ with a small limits and colimits, if we form the category of pointed objects ,i.e. the category with objects as morphisms ...
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Motivation behind the definition of monomorphism and epimorphism in category theory

Let $\mathcal{C}$ be a (locally small) category and let $X$ and $X'$ be objects in $\mathcal{C}$. (Definition $1$) We say that $f\colon X\to X'$ is a monomorphism if for any object $Y$ in ...
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Regarding constructing the $I$-adic completion of a ring

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For $n \geq m$, let $\varphi_{m,n}$ denote the canonical ring homomorphism $R / I^n \to R / I^m$. Let $J = \cap_n I^n$. Then $R / J$ ...
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Derived version of projection formula

Let $f \colon X \to Y$ be a continuous map of locally compact spaces. Denote by $Sh(X)$, $Sh(Y)$ the categories of sheaves of $k$-vector spaces for some field $k$ and by $D^b(X)$, $D^b(Y)$ their ...
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(co)Limits of naturally isomorphic functors

Let $F,G$ be naturally isomorphic functors from $\mathsf C$ to $\mathsf D$. How are the limits of colimits of these functors related? Does the existence of one guarantee the existence of the other?
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$\hom (M, \coprod_i N_i) \cong \bigoplus_i \hom (M, N_i)$ in abelian categories when $M$ is simple?

For $M$ a simple $R$ module, and $N_i$ a family of $R$ modules, we have $$ \hom (M, \bigoplus_i N_i) \cong \bigoplus_i \hom (M, N_i) $$ as abelian groups. Since the direct sum is the coproduct in the ...
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Group as a Category with One Object

A typical example in category theory is to consider a given group $G$ as a category with one object. Then, $\hom(G, G)$, the set of arrows from $G$ to $G$, is defined to be the elements of $G$. My ...
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Category theory and complexity classes

Is there any interesting way to make the set of computational complexity classes into a category? Almost every interesting mathematical class of objects forms a category after all.
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Natural morphism of sheaves $pr_1^{-1} F \otimes pr_2^{-1}G \rightarrow j_*(F\otimes G)$

I am reading a book, and the book said there is a natural map, which I don't know how. Can someone help me please? I can try to define the map, but it will be complicated, using sheafification many ...
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Formal definition of forgetful functor

Given the definition of a category $\mathbb{C}$ (that I rewrite just to agree on the notation), that consists of a collection of objects $\mathsf{Obj} ( \mathbb{C} )$; a collection of $\mathsf{Arr} ...
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A Question on a claim regarding the notion of “space” in “Indiscrete Thoughts”

I'm reading Gian-Carlo Rota's book "Indiscrete Thoughts". In page 220 I came across a strange quotation with very few explanations: We thought that the generalizations of the notion of space had ...
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Coproducts and pushouts of Boolean algebras and Heyting algebras

I am having trouble of find a reference explaining how to compute coproduts and pushouts in the category of Boolean algebras and in the category of Heyting algebras. To be precise I am looking for ...
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The functor $\mathbf{D} \rightarrow \mathbf{Prof}$ obtained by “splitting” $F : \mathbf{C} \rightarrow \mathbf{D}$ at each object of $\mathbf{D}.$

(Write $\mathbf{Prof}$ for the category whose objects are categories and whose arrows are profunctors.) I'm pretty sure that every functor $F : \mathbf{C} \rightarrow \mathbf{D}$ yields a ...
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121 views

Why do these equivalent categories seem to behave differently?

Write $\{1\}$ for the terminal category, and let $\Omega$ denote a category with two distinct (but isomorphic objects), call them $1_\Omega$ and $0_\Omega$, and a total of four arrows; two identities, ...
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Higher transformations between natural transformations and so on

In category theory, arrows between categories are functors, arrows between functors are natural transformations, so a natural question is to ask what are arrows between natural transformations ? I've ...
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Quotient of a category by equality in Grothendieck group

I'm currently studying a question in the area of categorification. The situation is that we have an abelian category $C$ and endofunctors $F, G$ on $C$ we really expected to be naturally isomorphic. ...
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51 views

What are the projective and the injective objects in the category of spectra?

What are the projective and the injective objects in the category of spectra (of simplicial sets)? Does the category of spectra have enough projectives and injectives? An object $P$ of a ...
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Does zero-kernel imply monic in Abelian categories?

I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ...
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CoKleisli category of the induced comonad of a monad

Given a monad, it induces a comonad on its Eilenberg-Moore category. We can then take the coKleisli category of this comonad. Can we say anything interesting about this?
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Lost Chevalley Manuscript

In the end notes of chapter 9 in Mac Lane's "Categories for the Working Mathematician", he mentions a lost Chevalley manuscript A systematic treatment of all possible properties of limits was ...
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Free objects; the way they are defined in terms of commutative diagrams

Wikipedia defines free objects as follows: Let $(\mathcal{C},F)$ be a concrete category (i.e. $F : \mathcal{C} \to {\rm \bf{Set}}$ is a faithful functor), let $X$ be a set (called basis), $A \in ...
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Categories enriched over a field

Let $K$ be a field, regarded as a monoidal category in the following way: Objects are elements $x\in K$; Morphisms between $x\to y$ are only identities; The (strictly symmetric) monoidal structure ...
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Full subcategories of $\mathsf{Grp}$ with epis/monos which are not surjective/injective

I read that there exist full subcategories of $\mathsf{Grp}$ in which there are epis which are not surjective, and ones in which there are monos that are not injective. I'm confused because they're ...
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Awodey's first UMP example

I am reading Awodey's "Category Theory" by myself and got stuck in a simple passage. He writes: If $g:A^\ast\rightarrow N$ satisfies $g(a)=f(a)$ for all $a\in A$ then, for all ...
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Why isn't the use category theory for graph transformation more prominent?

On the surface, it would seem that category theory would be a very natural and useful mathematical tool to address the subject of graph transformation. Yet, early indications from online searches seem ...