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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The classifying space of open covers of a manifold

Let $M$ be a manifold of dimension $d$ and let $\mathsf{Disk}_{/M}$ be the category of open subsets of $M$ that are diffeomorphic to $\mathbb{R}^d$ with morphisms given by inclusions. Let $\mathrm{B} ...
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1answer
34 views

Exponentiation category theory

Let be $$curry(eval_{AB})=id_{B^{A}}, $$ $B^{A}$ is an exponential object, $curry(g)(c)=g_{c}$ and $eval(f,a)=f(a)$, but how to prove it? Let $C$ be a category with all binary products and let $A$ ...
5
votes
1answer
130 views

Proposition 4.1 in Vistoli's notes on descent

Proposition 4.1 of Vistoli's notes says $\mathsf{Top}^2$ is a stack over $\mathsf{Top}$. The proof starts by looking at the fibered product $\coprod_iU_i \times _U\coprod_iU_i$, however this object ...
0
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1answer
46 views

Cartesian closed category isomorphism

Let be $$(B^{A})^{A'} \simeq B^{A \times A'}.$$ In any Cartesian closed category there is always an isomorphism but how to prove it?
3
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48 views

Extra hypotheses in proposition in Sheaves in Geometry and Logic?

In chapter I, section 9, proposition 5 in Mac Lane and Moerdijk's Sheaves in Geometry and Logic, it is stated that if $f : B' \to B$ is a morphism in a complete category $\mathcal{C}$ and the category ...
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111 views

Is this property of continuous functions equivalent to anything familiar? If not, does it at least have a name?

If I understand correctly, every morphism of topological spaces $f : Y \leftarrow X$ factorizes uniquely into a composite of three morphisms $$f = c \circ b \circ a$$ such that $c : Y \leftarrow ...
2
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1answer
47 views

Coherence diagrams for monoidal categories which have underlying sets are “automatically” natural?

I am just getting into monoidal categories, and first I am verifying at $FdVect$ is one, the monoidal operation being the usual vector-space tensor product. (Which I also just learned, so I may be ...
7
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2answers
90 views

Is this functor representable?

Fix a group $G_0$ and $R$ a subset of $G_0$. Consider the functor $F$ from $\textbf{Grps}$ to $\textbf{Sets}$, sending every object $G$ in $\textbf{Grps}$ to $F(G)$, the subset of $\varphi \in ...
1
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1answer
61 views

Is the category of simple graphs finitely complete?

I have read (on nLab and wikipedia) three conflicting statements: The category of simple graphs is finitely complete The category of simple graphs has no terminal object A category is finitely ...
2
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2answers
61 views

Limits by Products and Equalizers

The category $Cat$ of small categories is complete. Could you spell out with details the construction of the limit of a functor $F : J \to Cat$ by products and equalizers? (Mac Lane, Categories for ...
3
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1answer
42 views

Proving a category is *not* cartesian closed?

Are there any general techniques for proving a category is not cartesian closed? Do cartesian closed categories have nice, easily checked properties that finitely bicomplete distributive categories ...
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2answers
71 views

Proving some functor is adjoint to another. What to do with naturality condition?

Whenever I want to prove that some functor is (left/right) adjoint to another, I (mostly using hom-set definition) go on smoothly to prove the "isomorphism of the corresponding hom-sets", until it ...
1
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1answer
23 views

Extension of natural transformation from a dense subcategory

I'm trying to prove that the free cocompletion of a small category $\mathcal{C}$ gives an equivalence of categories $$Cat_+[\widehat{\mathcal{C}},\mathcal{D}] \longrightarrow ...
0
votes
0answers
49 views

Silly question about descent

Most sources say descent is defining an object over $S$ using objects over $U_i$ for some cover $\left\{ U_i \right\}$ of $S$. If I replace the covering family with a single arrow $\coprod _i ...
2
votes
1answer
58 views

Is the pullback of an adjunction along any functor an adjunction?

In $Cat$: If I take the pullback of a functor which has an adjoint along any functor, does the resulting functor have an adjoint of the same type (left-right)? If yes, does this adjoint behave nicely ...
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0answers
25 views

Commutation of pullbacks with disjoint unions in sites?

From what I understand, in superextensive sites disjoint unions commute with pullbacks. I know this holds in every topos sense pullback has a right adjoint, and I have heard something along the lines ...
2
votes
1answer
55 views

Group actions as colimit of torsors?

I'm learning group theory and I know a little bit about category theory(Mac Lane ch1-3,but have not appreciated what a colimit is). I know a group action can be viewed as a functor from a group as ...
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0answers
33 views

$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
2
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0answers
47 views

Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker ...
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0answers
49 views

Resources for learning functional programming/Haskell for the mathematically inclined.

I am a math student wanting to learn some functional programming with Haskell. From what I understand, many type theory concepts are analogous, even equivalent, to category theory concepts (e.g. ...
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0answers
45 views

Is it “okay practice” to exclude the epicness from the definition of extremal / strong epis?

Basically I wonder, whether I "should" include the property of being epi in the definition of extremal epis / strong epis (/...) (dually for extremal monos etc.). One hand it is terminology-wise a ...
2
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1answer
46 views

A question about colimits in enriched categories

I am just starting to learn about enriched categories, so excuse me if I am asking something trivial. Suppose $\mathcal{C}$ is a $\mathcal{V}$-enriched category $\mathcal{C}$, with $\mathcal{V}$ very ...
2
votes
1answer
60 views

Category theoretic definition of topological spaces.

We know that for algebraic structures like groups and rings, the axioms in their definition can be written in terms of objects and morphisms in the category of sets and hence can be generalised to ...
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0answers
41 views

Establish canonical isomorphism $Z^{Y \times X} \cong (Z^Y)^X$ for objects $X, Y$ and $Z$ from $\mathcal{AB}$ category of Abelian groups

Let $X^Y := \mathsf{Hom}_{\mathcal{AB}}(Y, X)$ be a set of all morphisms from objects $Y$ to $X$ from Abelian groups category $\mathcal{AB}$. Let $X \times Y$ be a product and $X + Y$ be a coproduct ...
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1answer
32 views

Free monoidal category over a set

From nLab's article on coherence theorems, there seems to be a notion of free monoidal category over a set $S$. I guess this corresponds to the left adjoint to the functor $Ob : MonCat \to Set$ which ...
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1answer
44 views

Homotopy of chain complexes (category theoretic proof)

I know the usual proof of the fact that if a morphism between chain complexes $f$ is homotopic to zero then it induces the $0$ map on cohomology. I was wondering if there is an easy proof of this ...
1
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1answer
53 views

Geometric xample of a one object cover which does not satisfy the sheaf condition?

For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self ...
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0answers
37 views

Limit-preserving functor that factors through representable functor.

This should be easy, but for some reason I'm not figuring it out right now. Suppose you have a locally small, complete category $\mathcal{C}$ and a functor $\mathcal{B} \xrightarrow{F} \mathcal{C}$. ...
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2answers
32 views

A trivial slice category

I am clumsily trying to read my way through Algebra: Chapter 0. Among the first examples of categories presented in the text is the following. Let $\mathcal{C}$ be a category and fix one of its ...
2
votes
1answer
42 views

Natural way of looking at projective transformations.

Let $k$ be a field and let $V$ and $W$ be finite-dimensional $k$-vector spaces, where $\dim(V)\ge1$ and $\dim(W)\ge1$. Let $q:V\to\mathbb{P}(V)$, $u\to[u]$ be the quotient map. By my teacher, a map ...
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1answer
32 views

how do contravariant 2-functors preserve adjunctions?

I know that covariant 2-functors preserve adjunctions. Do contravariant 2-functors preserve the order left-right of the adjoints, or do they reverse it?
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Categories of defintion for sites on spaces and sites on schemes

Starting with example 2.26 in Vistoli's Notes he defines topological sites on $\mathsf{Top}$, while sites on schemes are always defined on a slice category $\mathsf{Sch}/X$. Why is this?
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1answer
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Are representables on the étale site on topological space sheaves?

The gluing lemma says representables on the classical site on $\mathsf{Top}$ are sheaves. Basic scheme theory says the same is true for the small Zariski site of a scheme. Are representables on the ...
2
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2answers
42 views

Dividing left adjoints

If I have categories $C,D,E$ with "forgetful functors" $G_1:D\to C$ and $G_2:E\to D$, with $G_3=G_1\circ G_2:E\to C$, and left adjoints $F_1:C\to D$ and $F_3:C\to E$, is it possible to deduce a left ...
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3answers
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Group theoretic meaning of natural isomorphisms between certain functors

So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to $G$ specified by $\forall g$ $\in G$ $\exists$ $\phi(1)=g$. Each of these homomorphisms is in ...
2
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1answer
55 views

Is isomorphism defined between large categories?

By definition, an isomorphism between two objects in a category is a morphism so and so.. We know that $\mathbf {Cat}$ is the categories of small categories so that morphisms between objects are ...
4
votes
2answers
59 views

Why are invertible objects reflexive in a tensor category?

I am reading Deligne and Milne's notes on Tannakian categories. I'm only just getting used to the idea of an abstract tensor category, and I've encountered a very believable statement that I'm ...
2
votes
1answer
87 views

Why this intuition about natural transformations corresponds to its formal definition?

Almost everywhere people introduce the notion of natural transformations between two functors $ F$, $ G$ : $ \textbf C \Rightarrow \textbf D$ by examples like what follows: This is the intuition ...
3
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0answers
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modularizing category theory

I have made the experience that proofs using category theory often look very elegant and short but when it comes down to verifying the details there is quite a list of commutativities etc. to check. A ...
4
votes
1answer
110 views

Sheaf cohomology via resolutions vs. derived categories

So I know that when introducing sheaf cohomology, there are two main approaches via derived categories, and a perhaps more "down to earth" method of resolving by acyclic, fine, soft, sheaves. I'm ...
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1answer
37 views

Maximal subcategory inside a multicategory

Let $\mathcal M$ be a multicategory. Let $C(\mathcal M)$ be a category consisting of all objects and all unary multimorphisms of $\mathcal M$. Is there a standard name for $C(\mathcal M)$?
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26 views

Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
4
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0answers
33 views

Connection between cobar construction of DG-coalgebra and cobar construction from monad

Given a monad $M:C\to C$ we can construct a cobar resolution from it directly as a functor $\Delta\to [C,C]$ Given a DG-coalgebra $(C,d)$ we can construct a cobar resolution $\Omega C$ of it as ...
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1answer
28 views

Composition of stable-pseudomonomorphisms

Terminology Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object (not necessarily additive!). A morphism $f\colon A\rightarrow B$ is a pseudomonomorphism iff ...
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Show that $R_{1} \times R_{2}$ is not the coproduct of $R_{1}$ and $R_{2}$ in $\mathcal{R}$

Let $\mathcal{R}$ denote the category of rings. Show that $R_{1} \times R_{2} \simeq R_{1} \oplus R_{2}$ is not the coproduct of $R_{1}$ and $R_{2}$ in $\mathcal{R}$. I know if $R_{1} \times R_{2}$ ...
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1answer
44 views

Defining natural isomorphism without the language of category theory

I was wondering, if it was possible to fully define the natural isomorphisms without the language of category theory, but only with that of set theory. I am not interested in natural transformations ...
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2answers
50 views

Categorical interpretation of equality type

Consider the Martin Lof type theory. It's know that: product type correspond to product of two obects; unit type correspond to terminal object; and so on. The equality type corresponds to some ...
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1answer
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elementary question concerning definition of sifted colimit

I am reading a proof (in Algebraic Theories by Adamek et al, Theorem 2.15) for the fact that sifted categories $\mathcal{D}$ are precisely those for which the diagonal functor $\Delta : \mathcal{D} ...
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1answer
69 views

Where to find about the category theoretic study of manifolds?

I'm looking for a resource about a category theoretic study of manifolds. What do you think is a good start? Hint: Not after very advanced resources. So no worries (indeed, preferred) if it's an ...
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1answer
38 views

Limits in a functor category do what precisely to morphisms?

So I know that if $\textbf{C}$ is a category and $\textbf{D}$ is a complete category then the functor category $\textbf{D}^\textbf{C}$ is complete and limits are "computed pointwise" in the sense that ...