6.4: Decorated cospans

The goal of this section is to show how we can construct a hypergraph category whose morphisms are electric circuits. Doing all this will tie up lots of loose ends: colimits, cospans, circuits, and hypergraph categories.

Symmetric monoidal functors

Rough Definition 6.68.

Let (C, I(_{C}), ⊗(_{C})) and (D, I(_{D}), ⊗(_{D})) be symmetric monoidal cate- gories. To specify a symmetric monoidal functor (F, φ) between them,

(i) one specifies a functor F : C → D;

(ii) one specifies a morphism φ(_{I}) : I(_{D}) → F(I(_{C})).

(iii) for each c(_{1}), c(_{2}) (in) Ob(C), one specifies a morphism

φ(_{c1,c2}) : F(c(_{1})) ⊗(_{D}) F(c(_{2})) → F(c(_{1}) ⊗(_{C}) c(_{2})),

natural in c(_{1}) and c(_{2}).

We call the various maps φ coherence maps.

We require the coherence maps to obey bookkeeping axioms that ensure they are well behaved with respect to the symmetric monoidal structures on C and D. If φ(_{I}) and φ(_{c1,c2}) are isomorphisms for all c(_{1}), c(_{2}), we say that (F, φ) is strong.

Example 6.69.

Consider the power set functor P: Set Set. It acts on objects by sending a set S (in) Set to its set of subsets P(S) := {R (subseteq) S}. It acts on morphisms by sending a function f : S T to the image map imf : P(S) → P(T), which maps R (subseteq) S to {f (r) | r (in) R} (subseteq) T.

Now consider the symmetric monoidal structure ({1}, ×) on Set from Example 4.49. To make P a symmetric monoidal functor, we need to specify a function φ(_{I}) : {1} → P({1}) and for all sets S and T, a functor φ(_{S,T}) : P(S) × P(T) → P(S × T). One possibility is to define φ(_{I})(1) to be the maximal subset {1} (subseteq) {1}, and given subsets A (subseteq) S and B (subseteq) T, to define φ(_{S,T})(A, B) to be the product subset A × B (subseteq) S × T. With these definitions, (P, φ) is a symmetric monoidal functor.

Exercise 6.70.

Check that the maps φ(_{S,T}) defined in Example 6.69 are natural in S and T. In other words, given f : S S′ and g: T T′, show that the diagram below commutes:

Decorated cospans

Now that we have briefly introduced symmetric monoidal functors, we return to the task at hand: constructing a hypergraph category of circuits. To do so, we introduce the method of decorated cospans.

Circuits have lots of internal structure, but they also have some external ports also called ‘terminals’ by which to interconnect them with others. Decorated cospans are ways of discussing exactly that: things with external ports and internal structure.

To see how this works, let us start with the following example circuit:

We might formally consider this as a graph on the set of four ports, where each edge is labeled by a type of circuit component (for example, the top edge would be labeled as a resistor of resistance 2Ω). For this circuit to be a morphism in some category, i.e. in order to allow for interconnection, we must equip the circuit with some notion of interface. We do this by marking the ports in the interface using functions from finite sets:

Let N be the set of nodes of the circuit. Here the finite sets A, B, and N are sets consisting of one, two, and four elements respectively, drawn as points, and the values of the functions A N and B N are indicated by the grey arrows. This forms a cospan in the category of finite sets, for which the apex set N has been decorated by our given circuit.

Suppose given another such decorated cospan with input B

Since the output of the first equals the input of the second (both are B), we can stick them together into a single diagram:

The composition is given by gluing the circuits along the identifications specified by B, resulting in the decorated cospan

We’ve seen this sort of gluing before when we defined composition of cospans in Definition 6.45. But now there’s this whole ‘decoration’ thing; our goal is to formalize it.

Definition 6.75.

Let C be a category with finite colimits, and (F, φ) : (C, +) → (Set, ×) be a symmetric monoidal functor. An F-decorated cospan is a pair consisting of a cospan (A stackrel{i}{ ightarrow} N stackrel{o}{leftarrow} B) in C together with an element s F(N).5 We call (F, φ) the decoration functor and s the decoration.

The intuition here is to use C = FinSet, and, for each object N (in) FinSet, the functor F assigns the set of all legal decorations on a set N of nodes. When you choose an F decorated cospan, you choose a set A of left-hand external ports, a set B of right-hand external ports, each of which maps to a set N of nodes, and you choose one of the available decorations on N nodes, taken from the set F(N).

So, in our electrical circuit case, the decoration functor F sends a finite set N to the set of circuit diagrams graphs whose edges are labeled by resistors, capacitors, etc.—that have N vertices. Our goal is still to be able to compose such diagrams; so how does that work exactly?

Basically one combines the way cospans are composed with the structures defining our decoration functor: namely F and φ.

Let ((A stackrel{f}{ ightarrow} N stackrel{g}{leftarrow} B),s) and ((B stackrel{h}{ ightarrow} P stackrel{k}{leftarrow} C), t) represent decorated cospans. Their composite is represented by the composite of the cospan (A stackrel{f}{ ightarrow} N stackrel{g}{leftarrow} B) and (B stackrel{h}{ ightarrow} P stackrel{k}{leftarrow} C), paired with the following element of F(N +(_{B}) P):

(Fleft(left[iota_{N}, iota_{P} ight] ight)left(varphi_{N, P}(s, t) ight)) (6.76)

That’s rather compact! We’ll unpack it, in a concrete case, in just a second. But let’s record a theorem first.

Theorem 6.77.

Given a category C with finite colimits and asymmetric monoidal functor (F, φ) : (C, +) → (Set, ×), there is a hypergraph category Cospan(_{F}) whose objects are the objects of C, and whose morphisms are equivalence classes of F-decorated cospans.

The symmetric monoidal and hypergraph structures are derived from those on Cospan(_{C}).

Exercise 6.78.

Suppose you’re worried that the notation Cospan(_{C}) looks like the notation Cospan(_{F}), even though they’re very different. An expert tells you “they’re not so different; one is a special case of the other. Just use the constant functor F(c) := {∗}.” What does the expert mean?

Electric circuits

In order to work with the above abstractions, we will get a bit more precise about the circuits example and then have a detailed look at how composition works in decorated cospan categories.

Let’s build some circuits. To begin, we’ll need to choose which components we want in our circuit. This is simply a matter of what’s in our electrical toolbox. Let’s say we’re carrying some lightbulbs, switches, batteries, and resistors of every possible resistance. That is, define a set

(C:={ ext { light, switch, battery }} sqcupleft{x Omega mid x in mathbb{R}^{+} ight})

To be clear, the Ω are just labels; the above set is isomorphic to {light, switch, battery}⊔ (mathbb{R})+. But we write C this way to remind us that it consists of circuit components. If we wanted, we could also add inductors, capacitors, and even elements connecting more than two ports, like transistors, but let’s keep things simple for now.

Given our set C, a C-circuit is just a graph (V, A, s, t), where s,t : A V are the source and target functions, together with a function l : A C labeling each edge with a certain circuit component from C.

For example, we might have the simple case of V = {1,2}, A = {e}, s(e) = 1, t(e) = 2 so e is an edge from 1 to 2 and l(e) = 3Ω. This represents a resistor with resistance 3Ω:

Note that in the formalism we have chosen, we have multiple ways to represent any circuit, as our representations explicitly choose directions for the edges. The above resistor could also be represented by the ‘reversed graph’, with data V = {1, 2}, A = {e}, s(e) = 2, t(e) = 1, and l(e) = 3F.

Exercise 6.79.

Write a tuple (V, A, s, t, l) that represents the circuit in Eq. (6.71). ♦

A decoration functor for circuits. We want C-circuits to be our decorations, so let’s use them to define a decoration functor as in Definition 6.75.

We’ll call the functor (Circ, ψ). We start by defining the functor part

Circ : (FinSet, +) → (Set, ×)

as follows. On objects, simply send a finite set V to the set of C-circuits:

Circ(V) := {(V, A, s, t, l) | where s,t : A V, l : E C}.

On morphisms, Circ sends a function f : V V′ to the function

Circ( f ) : Circ(V) → Circ(V′);

(V, A, s, t, l) (longmapsto) (V', A, (s ; f), (t ; f), l).

This defines a functor; let’s explore it a bit in an exercise.

Exercise 6.80.

To understand this functor better, let c (in) Circ((underline{4})) be the circuit

and let f :(underline{4}) → (underline{3}) be the function

Draw a picture of the circuit Circ( f )(c). ♦

We’re trying to get a decoration functor (Circ, ψ) and so far we have Circ. For the coherence maps ψ(_{V,V'}) for finite sets V, V′, we define

psi_{V, V^{prime}}: operatorname{Circ}(V) imes operatorname{Circ}left(V^{prime} ight) longrightarrow operatorname{Circ}left(V+V^{prime} ight)
left((V, A, s, t, ell),left(V^{prime}, A^{prime}, s^{prime}, t^{prime}, ell^{prime} ight) ight) longmapstoleft(V+V^{prime}, A+A^{prime}, s+s^{prime}, t+t^{prime},left[ell, ell^{prime} ight] ight)
end{array}) (6.81)

This is simpler than it may look: it takes a circuit on V and a circuit on V′, and just considers them together as a circuit on the disjoint union of vertices V + V′.

Exercise 6.82.

Suppose we have circuits

in Circ((underline{2})).

Use the definition of ψ(_{V,V'}) from (6.81) to figure out what 4-vertex circuit ψ(_{underline{2},underline{2}})(b, s) (in) Circ((underline{2}) + (underline{2})) = Circ((underline{4})) should be, and draw a picture. ♦

Open circuits using decorated cospans. From the above data, just a monoidal functor (Circ, ψ) : (FinSet, +) → (Set, ×), we can construct our promised hypergraph category of circuits!

Our notation for this category is Cospan(_{Circ}). Following Theorem 6.77, the objects of this category are the same as the objects of FinSet, just finite sets. We’ll reprise our notation from the introduction and Example 6.42, and draw these finite sets as collections of white circles ◦.

For example, we’ll represent the object 2 of Cospan(_{Circ}) as two white circles:

These white circles mark interface points of an open circuit.
More interesting than the objects, however, are the morphisms in Cospan(_{Circ}).

These are open circuits. By Theorem 6.77, a morphism (underline{m}) → (underline{n}) is a Circ-decorated cospan: that is, cospan (underline{m}) → (underline{p}) ← (underline{n}) together with an element c of Circ((underline{p})).

As an example, consider the cospan (underline{1} stackrel{i_{1}}{ ightarrow} underline{2} stackrel{i_{2}}{leftarrow} underline{1}) where i(_{1})(1) = 1 and i(_{2})(1) = 2, equipped with the battery element of Circ((underline{2})) connecting node 1 and node 2. We’ll depict this as follows:

Exercise 6.84.

Morphisms of Cospan(_{Circ}) are Circ-decorated cospans, as defined in Definition 6.75. This means (6.83) depicts a cospan together with a decoration, which is some C-circuit (V, A, s, t, l) (in) Circ((underline{2})). What is it? ♦

Let’s now see how the hypergraph operations in Cospan(_{Circ}) can be used to construct electric circuits.

Composition in Cospan(_{Circ}). First we’ll consider composition. Consider the following decorated cospan from (underline{1}) to (underline{1}):

Since this and the circuit in (6.83) are both morphisms (underline{1}) → (underline{1}), we may compose them to get another morphism (underline{1}) → (underline{1}). How do we do this? There are two parts: to get the new cospan, we simply compose the cospans of our two circuits, and to get the new decoration, we use the formula Circ([ι(_{N}) , ι(_{P})])(ψ(_{N,P})(s, t)) from (6.76). Again, this is rather compact! Let’s unpack it together.

We’ll start with the cospans. The cospans we wish to compose are

We simply ignore the decorations for now.) If we pushout over the common set 1 = {◦}, we obtain the pushout square

This means the composite cospan is

In the meantime, we already had you start us off unpacking the formula for the new decoration. You told us what the map ψ(_{underline{2},underline{2}}) does in Exercise 6.82. It takes the two decorations, both circuits in Circ((underline{2})), and turns them into the single, disjoint circuit

in Circ((underline{4})). So this is what the ψ(_{N,P})(s, t) part means. What does the [ι(_{N}) , ι(_{P})] mean? Recall this is the copairing of the pushout maps, as described in Examples 6.14 and 6.25. In our case, the relevant pushout square is given by (6.85), and [ι(_{N}) , ι(_{P})] is in fact the function f from Exercise 6.80!

This means the decoration on the composite cospan is

Putting this all together, the composite circuit is

Exercise 6.86.

Refer back to the example at the beginning of Section 6.4.2. In particular, consider the composition of circuits in Eq. (6.73). Express the two circuits in this diagram as morphisms in Cospan(_{Circ}), and compute their composite. Does it match the picture in Eq. (6.74)? ♦

Monoidal products in Cospan(_{Circ}). Monoidal products in Cospan(_{Circ}) are much sim- pler than composition. On objects, we again just work as in FinSet: we take the disjoint union of finite sets. Morphisms again have a cospan, and a decoration.

For cospans, we again just work in Cospan(_{FinSet}): given two cospans A M B and C N D, we take their coproduct cospan A + C M + N B + D. And for decorations, we use the map ψ(_{M,N}) : Circ(M) × Circ(N) → Circ(M + N). So, for example, suppose we want to take the monoidal product of the open circuits


The result is given by stacking them. In other words, their monoidal product is:

Easy, right?

We leave you to do two compositions of your own.

Exercise 6.88.

Write x for the open circuit in (6.87). Also define cospans η : 0 → 2 and η : 2 → 0 as follows:

where each of these are decorated by the empty circuit (1, Ø, !, !, !) (in) Circ((underline{1})).(^{6})

Compute the composite η ; x ; ε in Cospan(_{Circ}). This is a morphism (underline{0}) → (underline{0}); we call such things closed circuits. ♦

6.4: Decorated cospans

NBC Olympics is presenting record coverage of the U.S. Olympic Team Trials, totaling 85+ hours and more than 50 hours in primetime across NBC, NBCSN, Olympic Channel: Home of Team USA, and NBC Olympics digital platforms in the lead up to the Games of the XXXII Olympiad in Tokyo, Japan, which will take place July 23-Aug. 8.

NBC Olympics’ 85.25 hours of programming is the most ever for the Team Trials, topping the 76.5 hours of coverage prior to the Rio Olympics in 2016. Throughout the month of June, NBC Olympics will present 22 days of Trials coverage, including 13 nights in primetime on NBC, covering four sports — swimming, gymnastics, track & field, and diving, which are expected to feature some of the most anticipated stories and athletes leading into the Tokyo Olympics, including:

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  • Track & Field: the most decorated American woman in Olympic track & field history and six-time Olympic gold medalist Allyson Felix, who is aiming for her fifth Olympics appearance at age 35 current 200m world champion Noah Lyles, who is attempting to make his first Olympic team 2016 Olympic 400m hurdles gold medalist Dalilah Muhammad and member of the 4x400m relay world championship team Sydney McLaughlin. Track & field coverage runs from June 18-27, featuring six nights of primetime on NBC.
  • Diving: four-time Olympic medalist David Boudia, aiming for his fourth consecutive team 2016 Olympic silver medalist Michael Hixon 2019 world bronze medalist Delaney Schnell and 2021 and 2019 world silver medalist Sarah Bacon. Schnell and Bacon look to make their first Olympic team this summer. Coverage takes place June 8-13 and includes live coverage of the men’s and women’s springboard and platform finals on NBC.

NBC Olympics has already provided live coverage of the USA Wrestling Trials from Fort Worth, Texas U.S. Olympic Marathon Trials from Atlanta, Ga. and the U.S. Rowing Trials from Sarasota, Fla. and West Windsor, N.J. and the NBC Sports app will stream coverage of all televised events, as well as several digital-only events. Select coverage on NBCSN or Olympic Channel can also be streamed on Peacock. Coverage of the U.S. Paralympic Team Trials will be announced soon.

*Broadcasts that air at different times in each time zone can be streamed live during the ET broadcast.

Date Coverage Time (ET) & Network
6/24 Men Day 1 6:30pm, NBCSN [REPLAY]
Men Day 1: Apparatus Feed 6:30pm, [REPLAY]
6/25 Women Day 1 7:30pm, Olympic Channel [REPLAY]
7pm CT
7pm MT
8pm PT
Women Day 1: Apparatus Feed 7:30pm, [REPLAY]
6/26 Men Day 2 3pm, Olympic Channel [REPLAY]
Men Day 2: Apparatus Feed 3pm, [REPLAY]
6/27 Women Day 2 8pm ET, NBC [REPLAY]*
7pm CT
7pm MT
8pm PT
Women Day 2: Apparatus Feed 8pm, [REPLAY]

The television schedule for Track & Field Trials is below. All events can be streamed on and the NBC Sports App. Any coverage of Track & Field Trials on NBCSN can also be streamed on Peacock.

*Broadcasts that air at different times in each time zone can be streamed live during the ET broadcast.

Date Coverage Time (ET) & Network
6/18 Qualifying 7pm, NBCSN [REPLAY]
Finals 10pm, NBC [REPLAY]
6/19 Qualifying 8pm, NBCSN [REPLAY]
Finals: W 100m, W Discus 10pm, NBC [REPLAY]
6/20 Finals: W 100mH, M 100m, W 400m, M 400m 9pm, NBC [REPLAY]
6/21 Qualifying 7pm, NBCSN [REPLAY]
Finals: M 800m, W 1500m, W 5000m, M Pole Vault, M Javelin, M Triple Jump 8pm ET, NBC [REPLAY]*
7pm CT
7pm MT
8pm PT
6/24 Finals: W Shot Put, W Steeplechase 9pm, NBCSN [REPLAY]
6/25 Finals: M Discus, M Steeplechase 5pm, NBCSN [REPLAY]
6/26 Final: Women's 10,000m 1pm, Olympic Channel [REPLAY]
Finals: W Javelin, M 400mH, W 10,000m, W 200m, M 110mH 9pm ET, NBC [REPLAY]*
8pm CT
8pm MT
9pm PT
6/27 Final: Men's 5000m 1pm, Olympic Channel [REPLAY]
Finals: M High Jump, W 400mH, M 5000m, W 800m, M 1500m, M 200m 11:30pm, NBCSN [REPLAY]

The rest of the live streaming schedule, featuring exclusive digital coverage of several individual events, is below.

6.4: Decorated cospans

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Las pláticas son los Miércoles , en horario 13:00hrs tiempo de Ciudad de México. Zona horaria / Time zone

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A case study in double categories

We give an elementary introduction to some ideas in double category theory by concentrating on one particular example, the double category of rings with homomorphisms, and bimodules. Along the way we discover a number of “new” morphisms of rings, interesting in their own right. We also look at interesting double adjunctions in this context.

No previous knowledge of double categories is assumed, just some familiarity with categories, rings and modules.

Polynomials and the dynamics of data

One can imagine a database schema as a category and an instance or state of that database as a functor I: C-->Set the category of these is denoted C-Set. One can think of a data-migration functor, a way of moving data between schemas C and D, as a parametric right adjoint C-Set --> D-Set. In database speak, these are D-indexed "unions of conjunctive queries".

Scene change. The usual semiring of polynomials in one variable with cardinal coefficients, polynomials such as p = y^3 + 3y + 2, can be categorified to Poly, the category of polynomial functors, where + and x are the categorical coproduct and product. Composition of polynomials (p o q) gives a monoidal operation on this category for which the identity polynomial, y, is the unit. Ahman and Uustalu showed in 2016 that, up to isomorphism, the comonoids in (Poly, o, y) are precisely categories (!), and Garner sketched a proof in a recent video that bimodules between polynomial comonoids are parametric right adjoints between copresheaf categories. Recall that these are precisely the data-migration functors described above. In the talk, I will describe this circle of ideas.

I propose that in 2021 a great transition is upon us distances that were measured in days are now measured in zoom-hiccups. The speed of data migration—if that's indeed a valid way to model it—is much faster than ever, driving dominance into the hands of those who move data: roughly speaking, computerized processes. Researchers use biomimicry to formalize as many aspects of human intelligence as they can, much of which is then installed as automated software systems that run constantly. I call the automated speed-up of bio-inspired intellectual processes AI, and I'm not judging it as good or bad, but I do consider it immensely important. I propose that we as mathematicians have the ability to shape the course of AI. Mathematics becomes technology, and I believe we'll fare better if that technology is based on elegant principles rather than made ad-hoc. Polynomial functors are my entry point, and this talk can serve as an invitation to others to join in whatever capacity appeals to them. To respect the standards of academic talks, I will mainly restrict my discussion to mathematics and its applications, rather than to speculation.

60 Gazebo DIY Plans

One of the loveliest attractions in a garden, backyard or outdoor area is a gazebo. It’s whimsical, charming and lively. It becomes the focal point of a garden setting. It is where special moments take place: proposals, romantic dinners, weddings, and christenings.

A gazebo can come in different forms, shapes, and sizes. The classic gazebo is a circular or octagonal in the area and has a pointed roof. This has an open design to let light and air in. The design could be country, rustic, contemporary or modern but the way it is built remains the same.

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Whether you are new to making gazebos or you have tried one before, here are updated and very popular plans on how to make traditional, sometimes contemporary-style, and gazebos. You will find these plans loaded with information not just on the gazebo being built but also the materials you need, the tools you need to use and the different steps to take. We know that with these plans, you will be able to make your own gazebo to light up your backyard.

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This is an open-air gazebo plan that was built on a hill or an area higher than the usual and overlooking a water feature. The structure has a charming appeal and was obviously made with such care. The posts are huge with intricate braces that support the ceiling. The roof has a triangular-shaped design with a bit of a skirt flowing outward. The cupola has similar material from the roof.

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Truly a gazebo with a classic design! You will find this challenging and at the same time very interesting because this gazebo does not have a regular shape. It has an octagonal design and therefore you need to construct this side after side. First, this structure was not merely raised from the ground. The best side was chosen where there were no low lying trees that can affect the gazebo structure.

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This gazebo was constructed in the midst of a lovely garden making it the highlight of the area. This is a moderately difficult build and not for a beginner at all.

3) The Shaded Gazebo Design

What a lovely place to relax in! This is an airy gazebo that houses a few chairs and a comfy sofa with soft cushions ready to lull you to sleep. This gazebo was built on a raised wooden deck. The gazebo has an open design but with balusters built around the wall of the structure. The eight narrow posts hold the roof with delicately shaped braces on each side of the posts.

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The posts are tall, reinforced by metal and cement which has been buried deep into the ground. The roof is tall, airy and has a structured design. Every design element here is well-structured and seamless making this an ideal area for relaxation and for special outdoor events.

5) The Prefab Gazebo

This gazebo is a pre-fabricated model made by a company that manufactures outdoor structures. It has a rectangular design with a raised deck. The walls are open with posts that bear decorative braces on each side. The ceiling is high and is decorated with shingles. This can be easily assembled in just a few hours and with just simple hand tools like a screwdriver, hammer and a drill.

The advantages of using prefab are obvious. It is easy to assemble, it costs less, it won’t lose its color and beauty despite years of use and won’t fall victim to pests and molds. However, this material is light and may not be a good idea in places where there is high wind. You can purchase a prefab gazebo and have it delivered and set up in just a day.

6) Backyard Sanctuary Gazebo Plan

This is another classic gazebo design with an open-air plan that will let the breeze in and the hot air out from the cupola. The roof is charmingly lined with small shingles which may have been painstakingly attached by the builder. The walls are simple with lovely low-lying balusters. This gazebo is on an elevated deck and accessed by a short flight of stairs.

The ceiling beams have also received the TLC as these match the balusters below. This gazebo plan is huge and could be challenging for a beginner. It is a structure for special events such as wedding and proposals or simply a nice place to relax and just take the time off.

7) Octagonal Gazebo Plans

This is an octagonal gazebo design with a modern twist. The gazebo area is very small and it simply fits a small seating area and a foldable table. The most eye-catching feature of this gazebo is its roof lined with lovely wooden panels. Even the ridge caps were expertly designed to hold each wooden slat. The beams for the ceiling extend from the center of the roof to the outer portion of the structure.

The walls of the gazebo are open save for the lower part with a unique wooden paneling for privacy. The posts are not too tall but not too short as well. This gazebo was made on a pebbled part of a yard and alongside a wall or fence. A truly challenging design that a beginner will surely enjoy building.

8) Grill and Gazebo in One

Why not just build a gazebo for relaxing while you can create one to house your grill? This is a clever use of space. The gazebo houses a grill and a backyard dining area as well. Built on a cemented area in a yard or garden, the small gazebo has strong posts and ceilings and has only two walls. Each of the walls was made to hold a small space to hold on to drinks, food and a few ornamental plants.

Bar seats were in place on each outer face of the wall where guests can sit and dine. The tall, triangular roofing of this gazebo can shield the grill as well as the cook and diners from the sun and rain. The roof is made of decorative aluminum sheets so it’s very cost-efficient. This is an interesting and functional design that is quite an easy project even for someone new to woodworking.

9) The Redwood Gazebo

This redwood gazebo plan has eight walls. This is a classic build with expertly measured can cut wooden panels to create elevated flooring, balusters, and beams for the roof. It has an open-air design making it perfect for relaxation. The roof is high and has inner plywood sheathing before shingles were added.

Decorative lattice beams are found along the top of each side which is one of the highlights of this build. The structure is strong and needs weatherproofing to keep the sun and rain from eating the lovely redwood.

As you can see, the plans for this redwood gazebo were expertly made. The plans have a complete list of materials and tools, detailed step by step instruction on how to make this project and some tips, perfect for beginners.

10) Western Red Cedar Gazebo

What makes red cedar a good material for outdoor furniture and structures like this lovely gazebo? Red cedar is not just visually-appealing but is strong and will naturally repel insects. It has natural oils that protect it from moisture and is innately resistant to rotting and aging.

Now, this gazebo was made of red cedar with an elevated flooring, balustered walls, high ceilings and shingled roof. A cupola removes hot air that accumulates from the tall ceiling so you will remain cool as you sit and relax. This gazebo is a fairly easy design especially if you have complete materials and basic woodworking knowledge and skills. It could be a good starting point for more intricate projects in the future.

Ramsey was born in Lincoln, Nebraska, the son of Mary Jane (née Bennett) (1919–1978) and James Dudley "Jay" Ramsey (1916–1992), a decorated World War II pilot. [1] [2] He attended Okemos High School in Michigan. [3] In 1966, he graduated from Michigan State University (MSU) with a bachelor's degree in electrical engineering. Ramsey earned a master's degree in business administration from MSU in 1971. [4]

Ramsey joined the Navy in 1966, served as a Civil Engineer Corps officer in the Philippines for three years, and in an Atlanta reserve unit for an additional eight years. [5]

In 1989, Ramsey formed the Advanced Product Group, one of three companies that merged to become Access Graphics. He became president and chief executive officer of Access Graphics, a computer services company and a subsidiary of Lockheed Martin. [6]

In 1996, Access Graphics grossed over $1 billion, and Ramsey was named "Entrepreneur of the Year" by the Boulder Chamber of Commerce. [7] Immediately following the murder of his daughter he was "temporarily replaced so the company did not have to bother him about business matters as he grieved", according to Lockheed Martin spokesman Evan McCollum, but returned to his job within weeks. [8]

Ramsey, along with his wife Patsy and young son Burke, moved back to Atlanta shortly thereafter. [9] Access Graphics was later sold to General Electric in 1997. [10]

His net worth was reported at $6.4 million as of May 1, 1996, prior to his daughter's murder. In 2015, John told Barbara Walters in an interview that the death of JonBenét and the ensuing investigation and cost of the case had cost him the entire family fortune. He also claims that because of the notoriety of the case he now finds it very difficult to find work. [ citation needed ] [11] [12]

The murder of Ramsey's six-year-old daughter, JonBenét, was the only murder in Boulder, Colorado, in 1996. [13]

The Boulder police considered the possibility that an intruder had gotten into the house and committed the murder. The Ramseys appeared on national television to assert their innocence. [ citation needed ]

Statements were given to the media by John Ramsey's ex-wife, his brother, and his sister-in-law. They categorically denied that John Ramsey was, or ever had been, a child abuser. Further, Ramsey's elder son, John Andrew, and elder daughter, Melinda, told interviewers that their father had always been a loving and gentle parent. [ citation needed ] Linda Arndt, the detective first assigned to the case, stated in a deposition that she believed John Ramsey was responsible for the murder. [14]

Letter from the District Attorney Edit

On July 9, 2008, the Boulder County District Attorney's office announced that, as a result of newly developed DNA sampling and testing techniques known as Touch DNA analysis, the Ramsey family members were no longer considered suspects in the case. [15] [16] In light of the new DNA evidence, Boulder County District Attorney Mary Lacy gave a letter to John Ramsey that same day, officially apologizing to the Ramsey family:

This new scientific evidence convinces us . to state that we do not consider your immediate family, including you, your wife, Patsy, and your son, Burke, to be under any suspicion in the commission of this crime.

The match of Male DNA on two separate items of clothing worn by the victim at the time of the murder makes it clear to us that an unknown male handled these items. There is no innocent explanation for its incriminating presence at three sites on these two different items of clothing that JonBenét was wearing at the time of her murder. To the extent that we may have contributed in any way to the public perception that you might have been involved in this crime, I am deeply sorry. No innocent person should have to endure such an extensive trial in the court of public opinion, especially when public officials have not had sufficient evidence to initiate a trial in a court of law. We intend in the future to treat you as the victims of this crime, with the sympathy due you because of the horrific loss you suffered.

I am aware that there will be those who will choose to continue to differ with our conclusion. But DNA is very often the most reliable forensic evidence we can hope to find and we rely on it often to bring to justice those who have committed crimes. I am very comfortable that our conclusion that this evidence has vindicated your family is based firmly on all of the evidence.

Several defamation lawsuits have ensued since JonBenét's murder. L. Lin Wood [17] [18] [19] was the attorney for the Ramsey family, filing defamation claims on their behalf against St. Martin's Press, Time, Inc., The Fox News Channel, American Media, Inc., Star, The Globe, Court TV, and The New York Post.

John and Patsy Ramsey were sued in two separate defamation lawsuits arising from the publication of their book, The Death of Innocence. These suits were brought by two persons named in the book as having been investigated by Boulder police as suspects in JonBenét's murder. The Ramseys were defended in those lawsuits by Lin Wood and three other Atlanta attorneys, James C. Rawls, Eric P. Schroeder, and S. Derek Bauer, who obtained dismissal of both lawsuits, including an in-depth decision by U.S. District Court Judge Julie Carnes that "abundant evidence" pointed to an intruder who committed the murder. [20]

In 2004, Ramsey campaigned for a seat in Michigan's House of Representatives for the 105th district. [21] He received 24.3 percent of the vote in the Republican Party primary, finishing in second place to Kevin Elsenheimer. [22]

Ramsey married Lucinda Pasch in 1966. They had three children. The couple divorced in 1978. [23] His eldest daughter, Elizabeth, was killed in a car crash at age 22 in 1992. [24]

Ramsey married his second wife, Patricia (Patsy) Paugh, in 1980, with whom he had two children, Burke and JonBenét. Patsy died of ovarian cancer in 2006 at age 49.

After his wife's death, Ramsey met Beth Holloway, mother of missing Natalee Holloway. It was reported that the two began dating. [25] However, Ramsey played down their relationship, stating that they "developed a friendship of respect and admiration" out of common interests related to their children. [26]

Ramsey relocated to Moab, Utah, and met his third wife, Jan Rousseaux, in 2011. They later married and relocated to Michigan. [27]

  • Ramsey was portrayed by Ronny Cox in the 2000 miniseriesPerfect Murder, Perfect Town. [28]
  • The Ramseys were also portrayed in the 2001 South Park episode "Butters' Very Own Episode" In a 2011 interview, South Park creators Trey Parker and Matt Stone stated that they regretted how the Ramseys were portrayed in the episode. [29]
  • The Ramseys were also portrayed in two MADtv sketches, one parodying Hollywood Squares, with John portrayed by Michael McDonald and Patsy by Alex Borstein.

Mentioned by Joey Diaz on the Joe Rogan Experience episode 1523 ft Joey "CoCo" Diaz and Brian Redban.

Snack Video

Snack Video is a social network (similar to TikTok) that lets you create and watch a bunch of short videos that you can share with other users. The interface might remind you quite a bit of another popular Chinese app. All you have to do is to browse the app and find the content that interests you the most. It's that easy.

Snack Video's main screen includes a bunch of videos that have been uploaded by some of the most popular users on this social network. On the other hand, you can also segment the posts uploaded by people who are nearby or those you follow. In any case, you'll only see short videos in a vertical format, and in most cases, they're accompanied by a song or sound that you can also click.

Another main selling point of Snack Video is that it includes a super user-friendly editor that you can use to post your videos. All you have to do is to record a video or add it directly from your smartphone's gallery. Plus, you'll find it really easy to add the most popular songs to your video while also decorating it with eye-catching filters.

Snack Video is a really complete platform where thousands of users upload their short videos to share them with the rest of the community. Everything under a well-organized interface that lets you search by theme so you can find exactly what you're looking for.

Warranty Coverage

One thing that is important to note is that a lot of carpets don’t provide warranty coverage for stairs. Although it seems strange (after all, most houses and even some condos/apartments have stairs!), historically manufacturers excluded all stairs from the carpet’s warranty.

In recent years, however, more carpets are being offered with a warranty on stairs. If you are concerned about having the protection a warranty can offer, then make sure that the carpet you choose for your stairs does in fact cover stairs in the warranty.


Any INF file that installs one or more devices must have a Manufacturer section. An IHV/OEM-supplied INF file typically specifies only a single entry in this section. If multiple entries are specified, each entry must be on a separate line of the INF.

Using a %strkey%=models-section-name entry simplifies the localization of the INF file for the international market, as described in Creating International INF Files and the reference page for the INF Strings section.

If an INF file specifies one or more entries in the manufacturer-name format, each such entry implicitly specifies the name of the corresponding Models section elsewhere in the INF.

You can think of each system-supplied INF file's Manufacturer section as a table of contents, because this section sets up the installation of every manufacturer's device models for a device setup class. Each entry in an INF file's Manufacturer section specifies both an easily localizable %strkey% token for the name of a manufacturer and a unique-to-the-INF per-manufacturer Models section name.

The models-section-name entries in the Manufacturer section can be decorated to specify target operating system versions. Different INF Models sections can be specified for different versions of the operating system. The specified versions indicate operating system versions with which the INF Models sections is used. If no versions are specified, Windows uses a specified Models section for all versions of all operating systems.

For Windows XP to Windows 10, version 1511, the format of TargetOSVersion decoration is as follows:

Starting with Windows 10, version 1607 (Build 14310 and later), the format of the TargetOSVersion decoration is as follows:

Each field is defined as follows:

Specifies the target operating system is NT-based. Windows 2000 and later versions of Windows are all NT-based.

Identifies the hardware platform. If specified, this must be x86, ia64, amd64, arm, or arm64.

Prior to Windows Server 2003 SP1, if Architecture is not specified, the associated INF Models section can be used with any hardware platform.

Starting with Windows Server 2003 SP1, Architecture must be specified in INF Models sections names for non-x86 target operating system versions. Architecture is optional in INF Models section names for x86-based target operating system versions.

A number that represents the operating system's major version number. The following table defines the major version for the Windows operating system.

Windows version Major version
Windows 10 10
Windows Server 2012 R2 6
Windows 8.1 6
Windows Server 2012 6
Windows 8 6
Windows Server 2008 R2 6
Windows 7 6
Windows Server 2008 6
Windows Vista 6
Windows Server 2003 R2 5
Windows Server 2003 5
Windows XP 5
Windows 2000 5

A number that represents the operating system's minor version number. The following table defines the minor version for the Windows operating system.

Windows version Minor version
Windows 10 0
Windows Server 2012 R2 3
Windows 8.1 3
Windows Server 2012 2
Windows 8 2
Windows Server 2008 R2 1
Windows 7 1
Windows Server 2008 0
Windows Vista 0
Windows Server 2003 R2 2
Windows Server 2003 2
Windows XP 1
Windows 2000 0

A number that represents one of the VER_NT_xxxx flags defined in Winnt.h, such as the following:



0x0000003 (VER_NT_SERVER)

If a product type is specified, the INF file is used only if the operating system matches the specified product type. If the INF supports multiple product types for a single operating system version, multiple TargetOSVersion entries are required.

A number representing a combination of one or more of the VER_SUITE_xxxx flags defined in Winnt.h. These flags include the following:












If one or more suite mask values are specified, the INF is used only if the operating system matches all the specified product suites. If the INF supports multiple product suite combinations for a single operating system version, multiple TargetOSVersion entries are required.

A number that represents the minimum OS build number of the Windows 10 release to which the section is applicable, starting with build 14310 or later.

The build number is assumed to be relative to some specific OS major/minor version only, and may be reset for some future OS major/minor version. Any build number specified by the TargetOSVersion decoration is evaluated only when the OS major/minor version of the TargetOSVersion matches the current OS (or AltPlatformInfo) version exactly. If the current OS version is greater than the OS version specified by the TargetOSVersion decoration (OSMajorVersion,OSMinorVersion), the section is considered applicable regardless of the build number specified. Likewise, if the current OS version is less than the OS version specified by TargetOSVersion decoration, the section is not applicable.

If build number is supplied, the OS version and BuildNumber of the TargetOSVersion decoration must both be greater than the OS version and build number of the Windows 10 build 14310 where this decoration was first introduced. Earlier versions of the operating system without these changes (for example, Windows 10 build 10240) will not parse unknown decorations, so an attempt to target these earlier builds will actually prevent that OS from considering the decoration valid at all.

Important We highly recommend that you always decorate models-section-name entries in the Manufacturer and Models sections with platform extensions for target operating systems of Windows XP or later versions of Windows. For x86-based hardware platforms, you should avoid the use of the .nt platform extension and use .ntx86 instead.

If your INF contains Manufacturer section entries with decorations, it must also include INF Models sections with names that match the operating system decorations. For example, if an INF contains the following Manufacturer section:

%FooCorp%=FooMfg, NTx86. 0x80, NTamd64

Then the INF must also contain INF Models sections with the following names:

[FooMfg.NTx86. 0x80]

This name applies to the Data Center suite of Windows XP and later versions of Windows on x86-based hardware platforms.


This name applies to all product types and suites of Windows XP and later versions of Windows on x64-based hardware platforms.

During installation, Windows selects an INF Models section in the following way:

  1. If Windows is running in an x86-based version of the operating system (Windows XP or later versions) that includes the Data Center product suite, Windows selects the [FooMfg.NTx86. 0x80]Models section.
  2. If Windows is running in an x64-based version of the operating system (Windows XP or later versions) for any product suite, Windows selects the [FooMfg.NTamd64]Models section.

If the INF is intended for use with operating system versions earlier than Windows XP, it must also contain an undecorated Models section named [FooMfg].

If an INF supports multiple manufacturers, these rules must be followed for each manufacturer.

The following are additional examples of TargetOSVersion decorations:

%FooCorp% = FooMfg, NTx86

In this example, the resultant INF Models section name is [FooMfg.NTx86], and is applicable for any x86 version of the operating system (Windows XP or later).

%FooCorp% = FooMfg, NT.7.8

In this example, for version 7.8 and later of the operating system, the resultant INF Models section name is [FooMfg.NT.7.8]. For earlier versions of the operating system such as Windows XP, [FooMfg.NT] is used.

Setup's selection of which INF Models section to use is based on the following rules:

  • If the INF contains INF Models sections for several major or minor operating system version numbers, Windows uses the section with the highest version numbers that are not higher than the operating system version on which the installation is taking place.
  • If the INF Models sections that match the operating system version also include product type and/or product suite decorations, Windows selects the section that most closely matches the running operating system.

Suppose, for example, Windows is executing on Windows XP (version 5.1), without the Data Center product suite, and it finds the following entry in a Manufacturer section:

%FooCorp%=FooMfg, NT, NT.5, NT.5.5, NT. 0x80

In this case, Windows looks for an INF Models section named [FooMfg.NT.5]. Windows also uses the [FooMfg.NT.5] section if it is executing on a Datacenter version of Windows XP, because a specific version number takes precedence over the product type and suite mask.

If you want an INF to explicitly exclude a specific operating system version, product type, or suite, create an empty INF Models section. For example, an empty section named [FooMfg.NTx86.6.0] prohibits installation on x86-based operating system versions 6.0 and higher.

The Overlooked Black History of Memorial Day

N owadays, Memorial Day honors veterans of all wars, but its roots are in America’s deadliest conflict, the Civil War. Approximately 620,000 soldiers died, about two-thirds from disease.

The work of honoring the dead began right away all over the country, and several American towns claim to be the birthplace of Memorial Day. Researchers have traced the earliest annual commemoration to women who laid flowers on soldiers’ graves in the Civil War hospital town of Columbus, Miss., in April 1866. But historians like the Pulitzer Prize winner David Blight have tried to raise awareness of freed slaves who decorated soldiers’ graves a year earlier, to make sure their story gets told too.

According to Blight’s 2001 book Race and Reunion: The Civil War in American Memory, a commemoration organized by freed slaves and some white missionaries took place on May 1, 1865, in Charleston, S.C., at a former planters’ racetrack where Confederates held captured Union soldiers during the last year of the war. At least 257 prisoners died, many of disease, and were buried in unmarked graves, so black residents of Charleston decided to give them a proper burial.

In the approximately 10 days leading up to the event, roughly two dozen African American Charlestonians reorganized the graves into rows and built a 10-foot-tall white fence around them. An archway overhead spelled out “Martyrs of the Race Course” in black letters.

About 10,000 people, mostly black residents, participated in the May 1 tribute, according to coverage back then in the Charleston Daily Courier and the New York Tribune. Starting at 9 a.m., about 3,000 black schoolchildren paraded around the race track holding roses and singing the Union song “John Brown’s Body,” and were followed by adults representing aid societies for freed black men and women. Black pastors delivered sermons and led attendees in prayer and in the singing of spirituals, and there were picnics. James Redpath, the white director of freedman’s education in the region, organized about 30 speeches by Union officers, missionaries and black ministers. Participants sang patriotic songs like “America” and “We’ll Rally around the Flag” and “The Star-Spangled Banner.” In the afternoon, three white and black Union regiments marched around the graves and staged a drill.

The New York Tribune described the tribute as “a procession of friends and mourners as South Carolina and the United States never saw before.” The gravesites looked like a “one mass of flowers” and “the breeze wafted the sweet perfumes from them” and “tears of joy” were shed.

This tribute, “gave birth to an American tradition,” Blight wrote in Race and Reunion: “The war was over, and Memorial Day had been founded by African Americans in a ritual of remembrance and consecration.”

In 1996, Blight stumbled upon a New York Herald Tribune article detailing the tribute in a Harvard University archive &mdash but the origin story it told was not the Memorial Day history that many white people had wanted to tell, he argues.

About 50 years after the Civil War ended, someone at the United Daughters of the Confederacy asked the Ladies Memorial Association of Charleston to confirm that the May 1, 1865, tribute occurred, and received a reply from one S.C. Beckwith: “I regret that I was unable to gather any official information in answer to this.” Whether Beckwith actually knew about the tribute or not, Blight argues, the exchange illustrates “how white Charlestonians suppressed from memory this founding.” A 1937 book also incorrectly stated that James Redpath singlehandedly organized the tribute &mdash when in reality it was a group effort &mdash and that it took place on May 30, when it actually took place on May 1. That book also diminished the role of the African Americans involved by referring to them as “black hands which only knew that the dead they were honoring had raised them from a condition of servitude.”

The origin story that did stick involves an 1868 call from General John A. Logan, president of a Union Army veterans group, urging Americans to decorate the graves of the fallen with flowers on May 30 of that year. The ceremony that took place in Arlington National Cemetery that day has been considered the first official Memorial Day celebration. Memorial Day became a national holiday two decades later, in 1889, and it took a century before it was moved in 1968 to the last Monday of May, where it remains today. According to Blight, Hampton Park, named after Confederate General Wade Hampton, replaced the gravesite at the Martyrs of the Race Course, and the graves were reinterred in the 1880s at a national cemetery in Beaufort, S.C.

The fact that the freed slaves’ Memorial Day tribute is not as well remembered is emblematic of the struggle that would follow, as African Americans’ fight to be fully recognized for their contributions to American society continues to this day.

Candace Parker still got it, and she got her coach, too

Candace Parker’s own peers voted her the most overrated player in the WNBA in 2019.

That year, she posted career-low averages of 11.2 points, 6.4 rebounds, and 3.5 assists per game on 42/27/79 shooting splits. The Los Angeles Sparks still finished 22-12 and first in the Western Conference, but lost 3-0 in the semifinals of the playoffs, where she posted just 10.5 points, 6.0 rebounds, and 3.5 assists per contest. And, in all honesty, she knew that year was disappointing she admitted it herself.

How did she respond? In the 2020 Wubble, she averaged 14.7 points, 9.6 rebounds, 4.6 assists, 1.2 steals, and 1.2 blocks while shooting 51/40/73. She also won her first Defensive Player of the Year Award and recorded 22 points, 14 rebounds, five assists, and two blocks.

This year with her hometown Chicago Sky, she’s averaged 13.1 points, 8.6 rebounds, and 4.1 assists through seven appearances. The Sky are 8-7 and tied for second in the Eastern Conference, just 1.5 games behind the first-place Connecticut Sun. Last night, she had her best game on her new squad, posting a season-high 23 points, accompanied by 12 rebounds and six assists while shooting 9-for-16 from the floor and 3-for-4 from three. Moreover, the 92-72 victory over the New York Liberty was Chicago’s sixth straight win.

Still hitting turnaround fadeaways. Still knocking down three-pointers. Still getting putbacks. Still faking out defenders in the post.

Parker currently stands 224 points away from 6,000 , and she’ll become the 12th player in WNBA history to achieve the mark, with Tina Charles becoming the 11th earlier this season. The Sky have 17 games remaining, and Parker would need to average 13.2 points the rest of the way — more or less her current average — to reach that milestone this season. She’s also 38 rebounds away from 3,000, a total she’d become the ninth WNBA player to reach.

After Chicago’s victory over the Liberty, Parker put on her media hat and invaded the post-game presser in an attempt to try and (playfully) get rid of morning shootarounds. As Sky head coach James Wade awaited the next question, Parker stepped in with an inquiry of her own.

“Your team was flying in late last night, and there was no shootaround this morning. Do you think in any type of way that that may have contributed to the way that they played today?” she asked, as Wade laughed his ass off the entire time.

“I think it helped a little bit,” Wade acknowledged before interrupting himself by not being able to contain his laughter.

Watch the video: 2008 F-250 FRONT COVER PT2 (October 2021).