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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Examples for the fact that a pullback of an epimorphism is not necessarily an epimorphism.

I'm reading in Borceux' book Basic Category Theory about pullbacks. It turns out that the pullback of an epimorphism is not necessarily an epimorphism. On the linked page, Borceux gives a ...
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Enough projectives and $F$ preserves limits implies $G$ preserves epi's.

Exercise: Let $\mathcal{C}, \mathcal{D}$ be categories, $G : \mathcal{C} \to \mathcal{D}$ and $F : \mathcal{D} \to \mathcal{C}$ an adjunction $F \dashv G$. Suppose $\mathcal{D}$ has enough projectives ...
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Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Since when are free abelian groups constructed w.r.t maps? Isn't the set $S$ all that matters? I don't understand what ...
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post composition as a functor?

For every arrow $f$, we have a post composition functor $f_\ast$. However, one often uses the equality $(gf)_\ast=g_\ast\circ f_\ast$. I was wondering, what's the actual definition of this functor ...
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Modal extensions (operators) for monoidal (categorical) logics

There is nice generalization of first order logic to monoidal (categorical) logics http://www.springer.com/us/book/9783642128202 which has recently been applied extensively as replacement for deontic ...
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Existence of pullback of fiber bundles from abstract nonsense?

Let $\mathsf C$ be a superextensive site with products. A trivial fiber bundle is a bundle $\pi:E\rightarrow B$ which is isomorphic to $\pi_1:B\times F\rightarrow B$ in $\mathsf{C}/B$. Let $\mathcal ...
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1answer
80 views

The left adjoint to the forgetful functor $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ and Barr-Beck

Let $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ be the forgetful functor from $\mathbb{C}$-vector spaces to $\mathbb{R}$-vector spaces. I am trying to explicitly construct the left ...
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45 views

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
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How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left ...
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Explicit unit/counit of inverse image/direct image adjunction.

Is there a nice explicit description for the unit and counit of the inverse image/direct image adjunction $f^{-1} \dashv f_*$ between sheaves of rings (and in the version $f^* \dashv f_*$ for ...
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48 views

What is this structure involving a monad and a comonad?

Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can ...
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384 views

Why is the homotopy category actually important?

Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the ...
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1answer
61 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
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77 views

Functorial modifications of a topology

Let $S$ be a set. Then there are only two ways to attach functorialy a topology $\mathcal{T(S)}$ to it: The discrete and the trivial topology. Functorial means in this case that all maps $f \colon S ...
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1answer
19 views

G adjoint iff initial object in $(D\downarrow G)$

$\def\D{\mathcal{D}}\def\C{\mathcal{C}}$ Let $G: \C \rightarrow \D$ be a functor. For each $D \in \D$ define the category $(D \downarrow G)$ which has as objects pairs $(C,g)$ with $C \in \C$ and $g: ...
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1answer
30 views

Functor is of the form Set(-,A)

Let $F: Set^{op} \rightarrow Set$ be a functor such that for corresponding functor $\overline{F}: Set \rightarrow Set^{op}$ we have $\overline{F} \dashv F$. With corresponding functor I mean that $F$ ...
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103 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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121 views

Is there some kind of deep relationship between substitution and recursion?

Define $\mathbb{N}$ as the initial object in the following category: Objects. Sets $X$ equipped with a function $S : X \rightarrow X$ and an element $0:X$. Morphisms. Functions that preserves ...
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1answer
86 views

Monads and Monoids-as-categories

I'm trying to understand the definition of a monad as a monoid and to identify this structure in the implementation of monads in Hakell. One definition is that of a structure $(T,\eta,\mu)$ given by ...
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48 views

Weighted limits in the $Cat$-category of categories

What is a weighted limit in the $Cat$-category of categories, functors and natural transformations? I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
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49 views

adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
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60 views

The free cocompletion of a complete locally small category is complete

$\DeclareMathOperator{\colim}{colim}\newcommand{\cat}{\mathbf}\DeclareMathOperator{\Nat}{Nat}$I'm looking for a reference that talks about the free cocompletion $\hat{\cat C}$ of a (large) locally ...
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1answer
89 views

Category which is not a subcategory of a complete category

By Yoneda lemma every small category $\mathcal{C}$ can be embedded in the cocomplete (and complete) category $[C^{op} ,\textbf{Set}]$. Most examples I know of large categories which are not complete, ...
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Expressing the stack of sheaves with 1-limits

Zhen Lin's answer to the MSE question mentions that $\mathsf{Sh}(-)$ is a stack for the canonical topology on the site of open subsets of a space. One of the comments asks whether the stack descent ...
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18 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...
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39 views

Is there a noteworthy example of an internal congruence in a category without binary products?

An internal relation in a category can be defined as a pair of jointly monic morphisms $r_1,r_2 : R \to A$ or equivalently as one monomorphism $r : R\to A\times A$, as long as the product $A\times A$ ...
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1answer
46 views

Concrete examples of epimorphism in the category of sets and partial functions

A epimorphism is a morphism $f : A \rightarrow B$ that is right-cancellative in the sense that, for all morphisms $g1, g2 : B \rightarrow X $ is true that \begin{equation} g_1\: o\: f = g_2\: o\: ...
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38 views

Is there an extremal epi which is not epi, although binary products exist?

It is known, that in a category $\mathcal{A}$: strong epis are epi, if $\mathcal{A}$ has binary products extremal epis are epi, if $\mathcal{A}$ has equalizers all strong epis are extremal all ...
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1answer
90 views

How to define a weighted cone?

Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X ...
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40 views

Groupoid objects in the category of algebras

Can anyone give me some references where I could read about groupoid objects in the category of algebras? References about groupoid objects in other categories would also be welcome.
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1answer
46 views

“Spanning” a category

What does it mean for objects to "span" a category? I know what a span is but I'm not exactly sure what the phrase means. Does it just mean the category consists of the specified objects? An example ...
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1answer
77 views

What do you call such an object?

I would like to know if there is a name for an object $X$ in a (finitely complete and cocomplete) category $\mathcal{C}$ which has the following property: $X$ is non-empty and for every sub-object ...
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2answers
82 views

The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category ...
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1answer
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Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
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1answer
87 views

The bigger picture the Five Lemma fits into

The Five Lemma is a statement in category theory about certain conditions under which certain maps in exact sequences are isomorphisms. It has a few relatives like the 4 lemmas and maybe the Nine ...
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1answer
48 views

In which situations is it possible to construct a coproduct this way?

Let $\left(F,U;\eta,\varepsilon\right):\mathcal{\mathcal{X}\rightharpoonup A}$ denote an adjunction where $U$ is faithful and where $\mathcal{X}$ is a category that has coproducts. You can think of ...
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1answer
32 views

Proving cardinality of coproduct presentation is unique without choice?

The definition of an extensive category immediately implies that given two coproduct decompositions indexed by sets of equal cardinality, if the coproduct objects are isomorphic compatibly with their ...
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1answer
38 views

$F$ is an equivalence of categories implies that $F$ is fully faithful and essentially surjective

I read in wikipedia that: One can show that a functor $F : C → D$ yields an equivalence of categories if and only if $F$ is full, faithful and essnetially surjective. I'm trying to prove this but I ...
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1answer
44 views

Showing epis in $\mathbf{Grp}$ are surjective

I'm working through Maclane's Categories, and I got to exercise 5 of Section 5. In this question, the reader is asked to prove that all epis in $\mathbf{Grp}$ are surjective, and gives a sketch of ...
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1answer
63 views

What would an infinite dimensional projective space look like as a scheme?

In topology, we can construct $\mathbb{CP}^\infty$ as the direct limit of $\cdots\rightarrow \mathbb{CP}^n \rightarrow \mathbb{CP}^{n+1}\rightarrow \cdots$ with the embedding given by $[x_0: x_1: x_2: ...
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What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where ...
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1answer
85 views

Homology as categorification of Euler characteristic

I am trying to understand: "Thus, the homology of a manifold M can be seen, in a sense, as a categorification of its Euler characteristic: a more sophisticated and richly structured bearer of ...
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1answer
39 views

equivalence of definitions for connected objects?

For an extensive category, the following conditions are equivalent for an object $C$. The representable copresheaf of $C$ commutes with coproducts. The $C=X\amalg Y\implies X\text{ or }Y$ is $0$ and ...
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What is category theory useful for?

Okay so I understand what calculus, linear algebra, combinatorics and even topology try to answer, but why invent category theory? In wikipedia it says it is to formalize. As far as I can tell it sort ...
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Forgetful functor applied to a module

I try to find a left adjoint to the forgetful functor $U: R-Mod \longrightarrow Ab$. I considered a functor $F:Ab \longrightarrow R-Mod$ defined by $F(G)=Hom(U(R),G)$. I'm not so sure that in this ...
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Finitely generated projective modules form exact category

I am looking for a proof of the fact that the category of f.g. projective modules over some commutative ring is exact. Actually, I would already be happy to know why this category is closed under ...
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1answer
376 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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When is the localization of a commutative ring a finitely generated projective module?

Let $R$ be a commutative ring and $M$ an $R$-module. The tensor product $(-)\otimes M$ has a left adjoint $(-)\otimes M^\ast$ for $M^\ast =\mathsf{hom}(M,R)$ iff $M$ is finitely generated projective. ...
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The benefit of writing Banach space theory in categorical language!

I was wondering if there exists a special benefit of writing Banach space theory in categorical language? I mean does there arise a hint of the existence of a connection with other mathematical field ...
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Slick Definition of the Category of Cartesian Closed Categories

I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ...