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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65 views

Deformation retracts are closed under pushouts

I'm having trouble with the proof of Proposition 2.4.9 in Hovey's Model Categories. Proposition. Deformation retracts are closed under pushouts. Proof. Suppose we have a pushout diagram ...
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93 views

Matrix associated to a linear transformation

Today my linear algebra teacher explained what is the matrix associated to a linear transformation between two finitely generated $\mathbb{K}$-vector spaces. In particular, if we have $B = \{v_1, ...
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2answers
118 views

Categorification of geometry

I don't know if this idea is known, relevant or dumb, but I noticed that one could define abstract connectedness with groupoids. Let us forget about topology for a while, and let us think ...
5
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1answer
78 views

Epimorphisms of monoids

I know that in the category of groups, the epimorphisms are precisely the surjective homomorphisms. What about the category of monoids? One can easily see, that surjective homomorphisms are epic (even ...
3
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1answer
56 views

Projection morphisms of categorical product

We say that an object $X$ is the categorical product of $X_1$ and $X_2$ if there exist morphisms $\pi_1$ and $\pi_2$, called projection morphisms, such that for every object $Y$ with morphisms $f_1 : ...
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24 views

A question about morphisms in a Grothendieck topos

I am not very familiar with topos theory so please excuse me if this is completely trivial. Fix an object $K$ in a Grothendieck topos $\mathcal{G}$. Let $k:0\to K$ be the unique morphism from the ...
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2answers
81 views

What is meant by “A and B represent the same functor whence are isomorphic” in this solution?

While browsing some old questions I came across the following: tensor product of sheaves commutes with inverse image It seemed like something interesting was going on in the answer, but I don't quite ...
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1answer
83 views

Lawvere algebraic theories as presentation-invariant

We can read in a lot of papers, included Lawvere's PhD thesis, that algebraic theories are "an invariant notion of which the usual formalism with operations and equations may be regarded as a ...
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3answers
94 views

What is the motivation for direct product? (categories)

This is the definition of direct product: let $\{X_i\}$, $i\in I$, be a family of objects in category $C$. A product $(X; p_i)$ is an object $X$, together with morphisms $p_i: X\rightarrow X_i$, with ...
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2answers
234 views

stratification (typage) of logic and syntax at the same time: is such a dream feasible? [closed]

This post is more philosophical than formal, yet I think it's an important question. There's an idea I have for long times already that would consist, in some sense, in doing a "theory of theories". ...
4
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1answer
44 views

Characters of group scheme represented by Cartier dual

For a commutative group scheme $\pi \colon G \to S$ finite locally free over a base scheme $S$ we let $A := \pi_* \mathcal{O}_G$ and $A^* = \mathcal{Hom}_\mathcal{O_S}(A, \mathcal{O}_S)$. Then $A^*$ ...
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2answers
127 views

On the philosophy of Category Theory

I have been told by my professor that Category Theory is not just a language but is a shift in the way we think. As an example he pointed out that in category theory we do not worry about the objects ...
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28 views

A category is locally finitely presented if the relative purity and purity coincide

Let $A$ be a locally finitely presented additive category, $X$ an additive subcategory. A sequence $0\rightarrow A_1\rightarrow A_2\rightarrow A_3\rightarrow 0$ in $A$ is pure exact if it is ...
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1answer
33 views

$n$-skeleton and the category of finite simplicial complexes

Let $C$ be the category of finite simplicial complexes with simplicial maps. Then I want to define a functor $F : C \rightarrow C_n$, where $C_n$ is the subcategory of $C$ consisting of complexes up ...
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2answers
220 views

Groupoids more fundamental than categories, really?

I've skimmed through a survey by Thierry Coquand on univalent foundations. It is claimed that "groupoids are more fundamental than categories". And that categories can be seen as groupoids equipped ...
2
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1answer
38 views

Viewing the universal property of rings of fractions as a universal arrow

For a multiplicatively closed subset $S$ of $A$, we have a functor $S^{-1}: A-Mod \rightarrow S^{-1}A-Mod$. I am trying to understand this functor a little bit better and I was thinking about the ...
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0answers
24 views

Are there variations on definition of locally finite category?

Wikipedia, under Categorical Algebra, defines a category as locally finite if, for each morphism $m$, the number of factorizations $m = \prod^n m_i$ (with no $m_i$ an identity) is finite. So for ...
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0answers
139 views

Arrow kernel in category theory and generalized equivalence relation

let $F : \mathcal{C} \rightarrow \mathcal{D}$ be a functor from two categories. It looks like that there are various notions of kernels one could define for a functor. One could define the arrow ...
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48 views

Are split exact sequences exact in the opposite direction?

In an abelian category, let $$0\longrightarrow A \overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C\longrightarrow0$$ be a split short exact sequence with $\ell f=1_A,gr=1_C$. Is the sequence ...
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1answer
79 views

Is it possible to construct ZFC set theory inside category theory?

It's entirely possible I don't understand what I am talking about, but I know that ZFC stands as a good foundation for much of mathematics and that category theory stands as a good foundation for ...
2
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1answer
55 views

is the property of representability of a sheaf on the big etale site checkable on the small site?

Let $S$ be a scheme and $F$ a sheaf on $(\textbf{Sch}/S)_\text{etale}$, whose restriction to the small etale site $S_\text{etale}$ is representable (in fact in my case this restriction is ...
5
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1answer
79 views

Initial and Terminal Objects in the category of rigs

I want to determine the initial and terminal objects in $\mathsf{Rig}$, the category of rigs (provided that they exist). I can't really find a source, book or web, that deals with this category at ...
6
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1answer
72 views

Is this a category?

In Awodey's book Category Theory (second edition), on page 5 he asks if we have a category by taking "sets as objects and as arrows, those $f\, :\, A \rightarrow B$ such that for all $b \in B$, the ...
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3answers
293 views

Why is the ring of integers initial in Ring?

In Algebra Chapter 0 Aluffi states that the ring $\Bbb{Z}$ of integers with usual addition and multiplication is initial in the category Ring. That is for a ring $R$ with identity $1_{R}$ there is a ...
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40 views

When defining a Grothendieck pretopology,can we get away with less than the fibre product axiom?

$\newcommand\restr[2]{{\left.#1\right|_{#2}}}$ I'm fairly new to this whole area, so correct me if there are any technical errors in any of this. The base category for a classical sheaf is the ...
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2answers
44 views

How can i show that the the category of pointed sets and the category of sets and partial functions are not isomorphic?

Denote $Pf$ the category of sets and partial functions and $Set_*$ the category of pointed sets. i can't see why, i was trying some contradiction argument with the definition of isomorphism, maybe ...
3
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1answer
60 views

Examples of colimits in a category of categories

Can someone give an example of a colimit in a category of categories? In particular, it would be nice to see a well known category as a colimit of other well known categories. Describe the diagram ...
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33 views

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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1answer
148 views

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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1answer
33 views

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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1answer
94 views

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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1answer
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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1answer
44 views

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 ...
2
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1answer
42 views

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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21 views

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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1answer
72 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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1answer
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 ...
4
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1answer
84 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 $f+a=b$. ...
6
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1answer
189 views

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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1answer
45 views

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 ...
2
votes
1answer
72 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 ...
2
votes
1answer
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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2answers
165 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 ...
3
votes
0answers
99 views

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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153 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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1answer
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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68 views

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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1answer
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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1answer
28 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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1answer
52 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 $a \rightarrow b$. ...