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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Easiest way to see that $\mathcal{C}$ is cocomplete?

Let $\mathcal{C}$ be a category that has all coproducts and coequalizers. My question is, what is the easiest way to see that $\mathcal{C}$ is cocomplete?
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Monotone map on a Preset

Related to my previous question I ran across this in the same book and thought I would try to get some clarity. If $R$ and $S$ are both Presets with relation $\le$ then the function $f$ is a monotone ...
2
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0answers
40 views

Algebra structure on dual to coalgebra

I'm trying to prove the following theorem using braided diagrams: Let $(C,\Delta,\varepsilon)$ be a finite-dimensional coalgebra. There is an algebra structure on $C^*$ given by multiplication ...
5
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2answers
134 views

A proof that right adjoints preserve limits?

Assume that categories $\mathscr{B}$ and $\mathscr{C}$ have all limits of shape $\mathbf{J}$. Then there's a slick proof that if $G\colon \mathscr{C} \to \mathscr{B}$ is a right adjoint, $G \circ ...
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3answers
111 views

On the importance of natural transformations

In p. 18 of Categories for the working mathematician (2d ed.), Mac Lane remarks that ..."category" has been defined in order to define "functor" and "functor" has been defined in order to define ...
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1answer
57 views

Is the category of (pre)sheaves over a singleton isomorphic to the category of sets?

I'm wondering whether $\mathsf{PSh}(\{x\})$ or $\mathsf{Sh}(\{x\})$ are equivalent to the category of sets. For sheaves, since the value on an initial object is a terminal object, It seems like the ...
5
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1answer
107 views

In what kinds of categories is a monic epi an isomorphism?

Is there a general description of categories $\mathscr{C}$ in which all monic epis are actually isomorphisms? In general, monic epis need not be isomorphisms. For example, in the ...
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31 views

Name (reference?) for lax monoidal functors that are 'full, or surjective, on monoids'?

A lax monoidal functor $F$ takes monoids to monoids. Is there a name for a lax monoidal functor that is 'full' or surjective with respect to this property? In other words, the functor $F : \mathcal{C} ...
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1answer
57 views

Inverse limit of blow up

Suppose $X_{0} = X$ is a complex space of dimension 2 with divisor $p_{0} \in X_{0}$. We can construct the blow-up, $X_{1}$ of $X$, which comes with a blow-down map $X_{1} \to X_{0}$. Suppose that ...
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0answers
35 views

How to speak on limit of sequence categorically? [duplicate]

I was thinking on ways to define limit of a sequence (over the reals, or over a metric space, or even better, over a general topological space) using the categorical limit (final or inicial object of ...
4
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1answer
72 views

Order in writing composed morphisms

When we have a function $f: A \rightarrow B$ between two sets, and we want to explicit that we are applying it to some element $x \in A$, we write $f(x)$. After this, is natural to write $f(g(x))$ ...
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90 views

Simple question in “Sheaves in geometry and logic”

There's an argument I don't understand in "Sheaves in geometry and logic" by Mac Lane and Moerdijk, that seems a priori easy but I can't see it. Page 174, diagram (10) (involving the power ...
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1answer
49 views

Any epi with codomain $P$ is split implies $P$ is projective [duplicate]

I'm struggling to prove that if any epi with codomain $P$ splits, then $P$ is a projective object. The converse direction I proved by factoring the identity to give a right inverse of the pi. How can ...
3
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2answers
61 views

How to show that naturally isomorphic functors preserve the same limits?

I've seen a number of times versions of the claim that if $F$ is naturally isomorphic to $G$, then if $F$ preserves limits of shape $\mathbf{J}$ so does $G$. (Sometimes with "It is easy to show that ...
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1answer
100 views

Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$

I'm reading Borceux's Handbook of Categorical Algebra, vol I, section 2.13 on filtered colimits. The author starts by constructing a canonical map $$\varinjlim _C \varprojlim _D F(C,D)\rightarrow ...
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1answer
46 views

When the unit of a universal property is an isomorphism

Let $G \colon \mathbf B \to \mathbf A$ be a functor, and let $A \in \mathbf A$ be an object. A universal arrow from $A$ to $G$ can be described by an isomorphism of functors \begin{equation} \mathbf ...
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55 views

Relation between categorical operations (limits and co-limits)

Suppose I have a diagram $B \longleftarrow A \longrightarrow C$ in a category, and I execute a push-out operation and get $B \longrightarrow D \longleftarrow C$. If I execute a pull-back over $B ...
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51 views

What categorical property do these forgetful functors have in common?

Consider the following examples: The forgetful functor $U_1: \operatorname{Vect}_\mathbb{C} \to \operatorname{Vect}_\mathbb{R}$ The forgetful functor $U_2: \operatorname{Diff}^{\text{or}} \to ...
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1answer
70 views

Any deeper “duality” between non-zero-divisors and units of a ring?

I'm reading Aluffi's algebra book at the moment -- specifically, I'm on the introductory rings/modules chapter. I noticed two interesting pieces of information: in a (not necessarily commutative) ...
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1answer
38 views

Are there different combinatorial species with the same symmetry type?

First off: for my purposes, let $\sf B$ be the category of finite sets with bijections, and ${\sf B}_n$ the subcategory of sets with cardinality $n$, and define a combinatorial species to be a functor ...
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3answers
267 views

What is the name for a function whose codomain and domain are equal?

What do we call a function whose domain and co-domain are the same set? Edit: While i expressed my question in terms of functions, domains and codomains, i was actually interested in the most ...
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1answer
48 views

Explanation of proof in Representation Theory: A Homological Point of View

in the book Representation Theory: A Homological Point of View Proposition 3.1.18 Zimmerman proves that a cokernel is a colimit, but I can't understand his proof. He lets $\left((M_i)_{i \in ...
2
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1answer
120 views

Initial elements in Set and identity

By definition for every object there is at least one morphism - identity, so, there must be identity morphism for Set initial object - empty set. But no function can have empty set as codomain, so, ...
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2answers
171 views

Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

The Church/Turing Thesis that we all know and love asserts that every algorithmically computable function (in an informally characterized sense) is in fact recursive/Turing computable/lambda ...
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1answer
50 views

How to call a category with a single morphism between every two objects?

How to call a category where for every pair of objects $A, B$, there is a unique morphism $f\colon A\to B$? (A trivial category?)
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60 views

Interaction of functors and homology in abelian categories

I'm working on exercise 1.6.H.a) of Ravi Vakil's algebraic geometry course notes. I'm aware that a question was posted on the same topic before (Prove the FHHF theorem using as much abstract non-sense ...
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0answers
40 views

Specific case of tensor-hom adjunction

I'm currently working on a project, for which I need various bits of category theory which I've not seen much of before and do not know in detail, so I would like some confirmation (and probable ...
2
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1answer
43 views

Why is the “functor category” functor $(C,B)\mapsto B^{C}$ contravariant in $C$?

Good day everyone: I have been reading the book Categories for the Working Mathematicians and it is written that the functor category $B^{C}$ is itself a functor of the categories $B$ and $C$, ...
3
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1answer
43 views

Multicategories with out-arities

Basically, my question is: Why the emphasis on domains in the notion of multicategory? I will now give the formal framework to state it correctly. Passing from categories to multicategories ...
2
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1answer
54 views

Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$

Let $f:X\rightarrow Y$ be an arrow in $\mathsf{Set}$. $f$ induces three functors: $$\exists (f),\forall (f):\mathcal P(X)\rightarrow \mathcal P(Y),\; \mathsf I(f):\mathcal P(Y)\rightarrow \mathcal ...
16
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1answer
186 views

Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
3
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1answer
34 views

Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$

Let $P$ be a $\mathsf{Set}$-valued presheaf and let $f^\ast:\mathsf{PSh}(Y)\rightarrow \mathsf{PSh}(X)$ be the (topological) inverse image sheaf functor, defined on objects as the filtered colimit ...
3
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1answer
50 views

Does the concept of “cograph of a function” have natural generalisations / extensions?

First, definitions: The graph of a function $f : A \to B$ is a subset of $A \times B$, namely the set $\{(x,y) : x \in A, y \in B, f(x) = y\}$. The cograph of a function $f : A \to B$ is the ...
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2answers
105 views

Prove that the free group generated by two elements is a coproduct of the integers by itself in Grp

That's a problem from Algebra: Chapter $0$ by P. Aluffi, p.78, ex.5.6. One needs to prove that the group $F(\{x,y\})$ is a coproduct $\mathbb Z*\mathbb Z$ of $\mathbb Z$ by itself in the category ...
2
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1answer
28 views

Coproduct of rooted posets

The questions Currently, I'm working with "Category Theory" by Steve Awodey (which is quite readable, by the way). Their, on page 68 he gives an example for a coproduct, the coproduct in rooted ...
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1answer
73 views

Equal Categories

Let $\infty$ be the "category" of all categories, where the objects are categories and the morphisms are functors. I am trying to motivate the definition of equal categories by doing it as follows. ...
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7answers
358 views

Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
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0answers
34 views

Understanding Tabulation in Rel Category

Tabulation in Allegories is a structure that is defined over the diagram $A \rightarrow B$. In a concrete Allegory, namely Rel category (Category of Sets and Relations) this seems to be the ...
5
votes
1answer
103 views

The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
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3answers
399 views

Monoids in Category Theory

I don't have a strong math background (engineering math) so I am at a bit of a disadvantage here but I have been trying to learn the broad strokes of Category Theory to help get a fuller picture of ...
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1answer
69 views

A functor preserves a product of $A$ and $B$ iff $F(A \times B) \cong F(A) \times F(B)$?

Let $F \colon \mathbf A \to \mathbf B$ be a functor, and let $A, B \in \mathbf A$. Assume that there exists a product $A \times B$, with projections $p \colon A \times B \to A$ and $q \colon A \times ...
2
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1answer
20 views

Category with zero

From "An Introduction to Ring Theory", Paul Cohn: "Let $\mathcal{A}$ be any category and define $\mathcal{A}''$ as the category obtained from $A$ by adjoining one object $Z$with a single morphism ...
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Why is the trivial group a zero object for the category of groups, but the empty set isn't a zero object for the category of sets? [duplicate]

I understand that the zero ring can't be a zero object for the category of rings, because in that case the 'arrows' are ring homomorphisms which, by definition, but maintain the unit. But in the ...
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1answer
42 views

Distributivity of pullbacks

If we consider morphisms $A\rightarrow C\leftarrow B$ in a category $\mathcal{C}$, then we denote their pullback by $A\leftarrow A\times_{C}B\rightarrow B$. The question is the following: is it true ...
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How is a connection the same as Morphism between different pullbacks to the first infinitesimal neighbourhood

This question arose from the first pages of Delignes Equations Differentielles. There he defines a connection on a vector bundle $V$ on an complex analytic space $X$ as the following Data: For all ...
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1answer
85 views

Properties preserved under equivalence of categories

I would like to ask about properties that are preserved under equivalence of categories. To be more specific, is it true that equivalences preserve limits? Why?
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2answers
87 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
2
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2answers
56 views

How can you take the dual of a category whose objects are Sets?

Let's say I have a category with two objects A {1, 2} B {3} I have the following morphisms ...
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1answer
33 views

Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
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108 views

Is there a special name for functors from a category C to a subcategory of C?

Is there a special name for functors from a category $C$ to a subcategory $S$ of $C$ where $S \ne C$? I'm using $\ne$ pretty informally above. I'd be happy with anything that reasonably captures the ...