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Adjugate matrix

adjoint matrix

The matrix $\mathrm{adj}(A)$ of cofactors of a square matrix $A$ over a field. Let $M_{ij}$ denote the minor formed by deleting the $i$-th row and $j$ column of $A$. The cofactor $C_{ij} = (-1)^{i+j} \det M_{ij}$ and $\mathrm{adj} A$ has $C_{ji}$ in the $(i,j)$ position.

The adjugate satisfies $$ \mathrm{adj}(A) \cdot A = A \cdot \mathrm{adj}(A) = (\det A) I $$ where $I$ denotes the identity matrix.

The term adjoint matrix is frequently used, but suffers from confusion with the adjoint matrix with respect to a Hermitian form. The untransposed matrix of cofactors is also seen.

References

[1]	A.C. Aitken, "Determinants and matrices", Oliver and Boyd (1939) Zbl 65.1111.05 Zbl 0022.10005
[1]	Thomas Muir. A treatise on the theory of determinants. Dover Publications (1960) [1933]

Span

Span may refer to

• Linear hull, also called linear span or span
• Span (category theory)

Span (category theory)

A diagram in a category of the form $$ \begin{array}{ccccc} & & C & & \\ & f \swarrow & & \searrow g & \\ A & & & & B \end{array} $$

Two spans with arrows $(f,g)$ and $(f',g')$ are equivalent if for all $D,p,q$ the diagrams $$ \begin{array}{ccccc} & & C & & \\ & f \swarrow & & \searrow g & \\ A & & & & B \\ & p \searrow & & \swarrow q \\ & & D & & \\ \end{array} \ \ \text{and}\ \ \begin{array}{ccccc} & & C & & \\ & f' \swarrow & & \searrow g' & \\ A & & & & B \\ & p \searrow & & \swarrow q \\ & & D & & \\ \end{array} $$ either both commute or both do not commute.

A pushout is the colimit of a span.

References

[1]	S. MacLane, "Categories for the working mathematician" , Springer (1971). ISBN 0-387-98403-8

Standard construction

A concept in category theory. Other names are triple, monad and functor-algebra.

Let $\mathfrak{S}$ be a category. A standard construction is a functor $T:\mathfrak{S} \rightarrow \mathfrak{S}$ equipped with natural transformations $\eta:1\rightarrow T$ and $\mu:T^2\rightarrow T$ such that the following diagrams commute: $$ \begin{array}{ccc} T^3 Y & \stackrel{T\mu_Y}{\rightarrow} & T^2 Y \\ \mu_{TY}\downarrow& & \downarrow_Y \\ T^2 & \stackrel{T_y}{\rightarrow} & Y \end{array} $$ $$ \begin{array}{ccccc} TY & \stackrel{TY}{\rightarrow} & T^2Y & \stackrel{T_{\eta Y}}{\leftarrow} & TY \\ & 1\searrow & \downarrow\mu Y & \swarrow1 & \\ & & Y & & \\ \end{array} $$

The basic use of standard constructions in topology is in the construction of various classifying spaces and their algebraic analogues, the so-called bar-constructions.

References

[1]	J.M. Boardman, R.M. Vogt, "Homotopy invariant algebraic structures on topological spaces" , Springer (1973)
[2]	J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978)
[3]	J.P. May, "The geometry of iterated loop spaces" , Lect. notes in math. , 271 , Springer (1972)
[4]	S. MacLane, "Categories for the working mathematician" , Springer (1971)

Comments

The term "standard construction" was introduced by R. Godement [a1] for want of a better name for this concept. It is now entirely obsolete, having been generally superseded by "monad" (although a minority of authors still use the term "triple" ). Monads have many other uses besides the one mentioned above, for example in the categorical approach to universal algebra (see [a2], [a3]).

References

[a1]	R. Godement, "Théorie des faisceaux" , Hermann (1958)
[a2]	E.G. Manes, "Algebraic theories" , Springer (1976)
[a3]	M. Barr, C. Wells, "Toposes, triples and theories" , Springer (1985)