Understanding ends as equalizers

While reading Fosco Loregian’s “This is the (co)end, my only (co)friend” I found the following remark: The product on the left ranges over objects in the category, but I’m confused by the so-called “double product” on the right, which seems to range over morphisms. In programmer-speak, that seems like a “type error” to me! Could … Read more

Is ⋂∞k=1Mk⋃∞k=1Nk\frac{\bigcap_{k=1}^\infty M_k}{\bigcup_{k=1}^\infty N_k} a direct limit?

Suppose we have a sequence of R-modules N1⊆N2⊆⋯⊆Nk⊆Nk+1⊆⋯⊆Mk+1⊆Mk⊆⋯⊆M2⊆M1. My question is: is it possible to make the following set {Pk:=MkNk|k≥1} with some ordering and with some R-modules homomorphsims to a directed set so that ⋂∞k=1Mk⋃∞k=1Nk is direct limit? Of course, if all Mk are equals to some fixed M, then for i,j∈N with i≤j(ordering w.r.t. … Read more

A monomorphism from a colimit

Let D be an upward directed poset, and suppose I have a diagram F:D→C in a cocomplete category such that F(d)→c is a monomorphism for all d∈D for some c∈C. Is it true that colimDF→c is also a monomorphism? When D is the empty category this is vacuously true. But in general, I don’t how … Read more

Adjoint functor with initial objects

I have to see that every left adjoint functor preserves initial objects. I prove it by Adjoint functor theorem which states that under certain conditions a functor that preserves colimits is a left adjoint. A basic result of the category theory is that left adjoint preserves all colimits, which can be characterized as initial objects. … Read more

Is there a simple way of visualising the direct limit of the cyclic subgroups of a group?

By way of background to this question, I am interested in the properties of direct limits. They are usually defined in terms that assume there is an underlying directed poset, but according to category theory, direct limits do in fact exist for general diagrams that are not directed. I am interested in getting an intuitive … Read more

Proving that the direct limit of a directed system is an equivalence relation.

From Dummit & Foote, pg. 268: Let I be an index set with a partial order. Suppose for every pair of indices i,j∈I with i≤j there is a map ρij:Ai→Aj such that the following hold: i. ρjk∘ρij=ρik 2. ρii=1 for all i∈I. Let B be the disjoint union of all the Ai. Define a relation … Read more

If one vertical arrow in a pullback is an iso, then so is the other

Consider the pullback square: \require{AMScd} \begin{CD} A @>f>> B\\ @Vg VV @VV h V\\ C @>>j > D \end{CD} Suppose h is an isomorphism. I’m trying to show that g is an isomorphism. I considered arrows 1_C,h^{-1}\circ j:C\to C and defined \alpha:C\to A as the unique map that makes the appropriate triangles commute. In particular … Read more

Direct limits and pullbacks

Suppose we have three directed sequences of C∗-algebras, say (An,φn),(Bn,ψn) and (Cn,θn) and ∗-homomorphisms αn:An→Cn and βn:Bn→Cn, then we can take the pullback An×CnBn for all n∈N and can also take the direct limit, thus lim. My question is: Does the following hold: \lim_{\rightarrow}{A_n\times_{C_n}B_n}=\lim_{\rightarrow}{A_n}\times_{\lim_{\rightarrow}{C_n}}\lim_{\rightarrow}{B_n} or in other words: do direct limits preserve pullbacks? From my … Read more

Definition of direct limit of groups by Serre

I’ve started to read the book Trees by Jean Pierre-Serre, and in Its first section he defines the direct limit of groups. I’m pretty used to the canon definition of direct limits which requires a directed set of indexes I and a set of morphisms satisfying certain conditions (see for example this). The Serre’s definition … Read more