science

Pseudo-functor

Pseudo-functor is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pseudo-functor rather than just read about it. In short: In mathematics, a pseudofunctor F is a mapping from a category to the category Cat of (small) categories that is just like a functor except that F ( f ∘ g ) = F ( f ) ∘ F ( g ) {\displaystyle F(f\circ g)=F(f)\circ F(g)} and F ( 1 ) = 1 {\displaystyle F(1)=1} do not hold as exact equalities but only up to coherent isomorphisms. A typical example is an assignment to each pullback F f = f ∗ {\displaystyle Ff=f^{*}} , w…

Key takeaways

Pseudo-functor belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pseudo-functor to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pseudo-functor from memory before moving on to harder problems.

Reference excerpt

In mathematics, a pseudofunctor F is a mapping from a category to the category Cat of (small) categories that is just like a functor except that F ( f ∘ g ) = F ( f ) ∘ F ( g ) {\displaystyle F(f\circ g)=F(f)\circ F(g)} and F ( 1 ) = 1 {\displaystyle F(1)=1} do not hold as exact equalities but only up to coherent isomorphisms. A typical example is an assignment to each pullback F f = f ∗ {\displaystyle Ff=f^{*}} , which is a contravariant pseudofunctor since, for example for a quasi-coherent sheaf F {\displaystyle {\mathcal {F}}} , we only have:

( g ∘ f ) ∗ F ≃ f ∗ g ∗ F . {\displaystyle (g\circ f)^{*}{\mathcal {F}}\simeq f^{*}g^{*}{\mathcal {F}}.}

Since Cat is a 2-category, more generally, one can also consider a pseudofunctor between 2-categories, where coherent isomorphisms are given as invertible 2-morphisms. The Grothendieck construction associates to a contravariant pseudofunctor a fibered category, and conversely, each fibered category is induced by some contravariant pseudofunctor. Because of this, a contravariant pseudofunctor, which is a category-valued presheaf, is often also called a prestack (a stack minus effective descent).

Definition A pseudofunctor F from a category C to Cat consists of the following data

a category F ( x ) {\displaystyle F(x)} for each object x in C, a functor F f {\displaystyle Ff} for each morphism f in C, a set of coherent isomorphisms for the identities and the compositions; namely, the invertible natural transformations

F ( f ∘ g ) ≃ F f ∘ F g {\displaystyle F(f\circ g)\simeq Ff\circ Fg} ,

F ( id x ) ≃ id F ( x ) {\displaystyle F(\operatorname {id} _{x})\simeq \operatorname {id} _{F(x)}} for each object x such that

F ( f g h ) → ∼ F ( f g ) F h → ∼ F f F g F h {\displaystyle F(fgh){\overset {\sim }{\to }}F(fg)Fh{\overset {\sim }{\to }}FfFgFh} is the same as F ( f g h ) → ∼ F f F ( g h ) → ∼ F f F g F h {\displaystyle F(fgh){\overset {\sim }{\to }}FfF(gh){\overset {\sim }{\to }}FfFgFh} ,

F ( id x ) ∘ F f → ∼ F ( id x ∘ f ) = F f {\displaystyle F(\operatorname {id} _{x})\circ Ff{\overset {\sim }{\to }}F(\operatorname {id} _{x}\circ f)=Ff} is the same as F ( id x ) ∘ F f ≃ id F ( x ) ∘ F f = F f {\displaystyle F(\operatorname {id} _{x})\circ Ff\simeq \operatorname {id} _{F(x)}\circ Ff=Ff} , and similarly for F f ∘ F ( id x ) {\displaystyle Ff\circ F(\operatorname {id} _{x})} .

Higher category interpretation The notion of a pseudofunctor is more efficiently handled in the language of higher category theory. Namely, given an ordinary category C, we have the functor category as the ∞-category

Fct ( C , Cat ) . {\displaystyle {\textbf {Fct}}(C,{\textbf {Cat}}).}

Each pseudofunctor C → Cat {\displaystyle C\to {\textbf {Cat}}} belongs to the above, roughly because in an ∞-category, a composition is only required to hold weakly, and conversely (since a 2-morphism is invertible).

Worked examples

Example 1 — a first encounter with Pseudo-functor

Start with the simplest possible case. Write down what Pseudo-functor claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pseudo-functor before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pseudo-functor ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pseudo-functor

In research: Pseudo-functor appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pseudo-functor in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pseudo-functor is common in secondary-school and first-year university syllabi. It links to neighbouring topics Category theory stubs, Functors, so understanding it makes those chapters shorter.
In everyday life: Look for Pseudo-functor outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pseudo-functor in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pseudo-functor means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pseudo-functor out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pseudo-functor in simple terms?

Why does Pseudo-functor matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pseudo-functor?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pseudo-functor.